Measuring the properties of nearly extremal black holes with gravitational waves
Katerina Chatziioannou, Geoffrey Lovelace, Michael Boyle, Matthew Giesler, Daniel A. Hemberger, Reza Katebi, Lawrence E. Kidder, Harald P. Pfeiffer, Mark A. Scheel, Béla Szilágyi
MMeasuring the properties of nearly extremal black holes with gravitational waves
Katerina Chatziioannou, Geoffrey Lovelace, Michael Boyle, Matthew Giesler, Daniel A. Hemberger, Reza Katebi,
2, 5
Lawrence E. Kidder, Harald P. Pfeiffer,
1, 6
Mark A. Scheel, and B´ela Szil´agyi Canadian Institute for Theoretical Astrophysics, 60 St. George Street, University of Toronto, Toronto, ON M5S 3H8, Canada Gravitational Wave Physics and Astronomy Center, California State University Fullerton, Fullerton, California 92834, USA Cornell Center for Astrophysics and Planetary Science, Cornell University, Ithaca, New York 14853, USA Theoretical Astrophysics 350-17, California Institute of Technology, Pasadena, CA 91125, USA Department of Physics and Astronomy, Ohio University, Athens, Ohio 45701, USA Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Am M¨uhlenberg 1, 14476 Potsdam-Golm, Germany (Dated: August 23, 2018)Characterizing the properties of black holes is one of the most important science objectives forgravitational-wave observations. Astrophysical evidence suggests that black holes that are nearlyextremal (i.e. spins near the theoretical upper limit) might exist and, thus, might be among themerging black holes observed with gravitational waves. In this paper, we explore how well currentgravitational wave parameter estimation methods can measure the spins of rapidly spinning blackholes in binaries. We simulate gravitational-wave signals using numerical-relativity waveforms fornearly-extremal, merging black holes. For simplicity, we confine our attention to binaries with spinsparallel or antiparallel with the orbital angular momentum. We find that recovering the holes’ nearlyextremal spins is challenging. When the spins are nearly extremal and parallel to each other, theresulting parameter estimates do recover spins that are large, though the recovered spin magnitudesare still significantly smaller than the true spin magnitudes. When the spins are nearly extremaland antiparallel to each other, the resulting parameter estimates recover the small effective spin butincorrectly estimate the individual spins as nearly zero. We study the effect of spin priors and arguethat a commonly used prior (uniform in spin magnitude and direction) hinders unbiased recovery oflarge black-hole spins.
I. INTRODUCTION
Beginning with the first discovery of gravitational waves(GWs) passing through Earth in 2015, to date the Laser In-terferometer Gravitational-Wave Observatory (LIGO) [1]and Virgo [2] have announced five detections of GWs frommerging binary black holes (BBH) [3–7]. As LIGO andVirgo approach their design sensitivity, they are expectedto detect hundreds of merging BH binaries [5, 8].One important objective of the GW observations isthe measurement of the masses and spins of the mergingBHs. This is interesting in its own right, but accuratecharacterization of the systems’ properties is also crucialfor astrophysical inference. The masses and the spins ofthe binary components can reveal information about theway these binaries were formed and about the propertiesof the BH progenitors. While most formation scenariospredict similar mass distributions for merging BHs [9–11], it has been suggested that spin measurements mightbe able to offer information about different formationchannels and the BH progenitor properties, e.g. [12–23].Besides spin directions, spin magnitudes carry impor-tant information as well, since they depend on the angularmomentum of the BH’s stellar progenitor and its evolution.At the moment, there remains considerable uncertainty inBH spin measurements, with mild tension between spinsinferred from GW observations [3–7], stellar evolutionmodels [24] and X-ray binary observations [25]. BH spinsinferred from GW observations to date have pointed to-wards slowly spinning BHs, while inferences of BH spinsfrom X-ray binaries tend to be higher, including some inferred spins that are nearly extremal [26, 27], thoughthese BHs need not be part of the same population [28].By nearly extremal, we mean spins close to the theoreti-cal maximum for a Kerr BH, i.e., dimensionless spins χ satisfying χ ≡ SM ≈ , (1)where S is the spin angular momentum and M is themass of the spinning BH and throughout the paper weuse units where G = c = 1.GW observations primarily provide information aboutthe effective spin χ eff , a combination of the spin compo-nents along the binary’s orbital angular momentum thatis conserved to second post-Newtonian order [29, 30].Specifically, χ eff = m ( (cid:126)χ · ˆ L ) + m ( (cid:126)χ · ˆ L ) m + m , (2)where m and m are the the masses of the larger andsmaller BH respectively, ˆ L is a unit vector in the directionof the orbital angular momentum, and (cid:126)χ and (cid:126)χ are thedimensionless spin vectors of the BHs. The apparent dis-crepancy between GW and X-ray binary measurementshas lead to stellar evolution models predicting a bimodal-ity in the spin distribution of BHs. These models suggest The second post-Newtonian order is a term of order ( v/c ) relativeto the leading-order term, where v is some characteristic velocityof the systems and c is the speed of light. a r X i v : . [ g r- q c ] A ug that some BHs in future LIGO observations might havelarge spins that are also aligned with the orbital angularmomentum [31, 32].In this paper we pose the following question: if the BHsin a LIGO source were to have nearly extremal spins, couldwe tell? To address this question, we simulate GW signalsusing numerical-relativity (NR) waveforms computed withthe Spectral Einstein Code (SpEC) [33]. We use twoSpEC simulations from the public Simulating eXtremeSpacetimes (SXS) catalog [34, 35] and two new, previouslyunpublished simulations, including one with the highestBH spins simulated to date. Three simulations haveBH spin magnitudes nearly extremal and spin directionseither both parallel to ˆ L , both antiparallel to ˆ L or onespin parallel and one antiparallel. The fourth simulationhas moderate BH spins and is included to help assessthe impact of large individual spin magnitudes when theypoint in opposite directions. We then use LIGO parameterestimation methods and tools to infer the properties ofthe simulated signals, including their masses and spins.We find that current parameter estimation methodscan recover large spins, but only if the effective spin islarge (meaning that the spins are either both aligned orboth antialigned with the orbital angular momentum) andthe signal-to-noise ratio (SNR) is sufficiently high. Therecovered spin magnitudes and effective spin are shiftedsignificantly towards less extremal values under the mostcommonly used spin prior assumption. If, on the otherhand, the effective spin is small (meaning that the twoBHs’ spins point in opposite directions), we accuratelyrecover the small effective spin but incorrectly recoversmall individual spins. Our results suggest that if theUniverse contains BBH systems with nearly extremalspins, GW inference might fail to tell us.The rest of this paper is organized as follows. In Sec. II,we describe the NR waveforms, and the simulated GWsignals that we generate from them, as well as our pa-rameter estimation methods. In Sec. III, we present ourresults and discuss the conditions under which we canmeasure large spins. We conclude in Sec. IV. II. METHODS
We calculate our simulated gravitational waveformsusing SpEC. SpEC’s methods, including recent improve-ments enabling more robust simulations of merging BHswith nearly extremal spins, are described in Ref. [36] andthe references therein.We consider four numerical gravitational waveformsfrom merging BHs, each simulated with SpEC. TheBHs in each simulation have spins either aligned or an-tialigned with the orbital angular momentum. Two ofthese simulations (SXS:BBH:0305 and SXS:BBH:0306)were previously presented in Refs. [3, 37, 38] and areavailable in the public SXS catalog [35], while the othertwo (SXS:BBH:1124 and SXS:BBH:1137) are new. Theconfigurations are summarized in Table I: SXS:BBH:1124
SXS:BBH: 1 /q χ z χ z χ eff N orbits f GW (Hz)0305 1.22 0.330 -0.439 -0.016 15.2 16.80306 1.3 0.961 -0.899 0.152 12.6 19.41124 1 0.998 0.998 0.998 25 14.21137 1 -0.969 -0.969 -0.969 12 14.9TABLE I. Properties of the SXS simulations used in this paper.The table shows the mass ratio q , spin χ of the larger BH,spin χ of the smaller BH, the resulting χ eff , the number oforbits N orbits in the simulation, and the initial GW frequencyof the ( (cid:96) = 2 , m = 2) mode for a system with a total mass of70 M (cid:12) . has large spins aligned with the orbital angular momen-tum; SXS:BBH:1137 has large spins antialigned with theorbital angular momentum; SXS:BBH:0306 has two largespin pointing in opposite directions, resulting in a small ef-fective spin; and SXS:BBH:0305 has moderate antiparallelspins and a small effective spin.We use the numerical-relativity (NR) data to simulateGW signals as observed by the two Advanced LIGOdetectors with the projected sensitivity for the secondobserving run [39]. As is common practice, we do not adddetector noise on the simulated signal, which is equivalentto averaging over noise realizations [40]. All intrinsicparameters of the simulated signals apart from the totalmass are determined by the NR data and are given inTable I. The total mass of the system is an overall scalefactor in vacuum general-relativity that we are free tospecify. We select extrinsic parameters such that theorbital angular momentum of the binary points towardsthe GW detectors and place the binary systems over theLivingston detector, scaling the source distance to achievea signal-to-noise ratio (SNR) of interest. See Ref. [44] fora description of the details and implementation of the NRinjection infrastructure we make use of.We then analyze the simulated data with the parame-ter estimation software library LALInference [45], whichsamples the joint multidimensional posterior distributionof the binary parameters. The posterior distribution iscalculated through Bayes’ Theorem p ( (cid:126)x | d ) ∼ p ( (cid:126)x ) p ( d | (cid:126)x ),where p ( (cid:126)x | d ) is the joint posterior for the parameters (cid:126)x given data d , p ( (cid:126)x ) is the prior distribution, and p ( d | (cid:126)x )is the likelihood for the data. In GW parameter estima-tion and under the assumption of stationary and Gaus-sian detector noise, the likelihood can be expressed asln p ( d | (cid:126)x ) ∼ − / d − h ( (cid:126)x ) | d − h ( (cid:126)x )), where parenthesesdenote the noise-weighted inner product [46] evaluatedfrom a lower frequency of 20Hz (24Hz for SXS:BBH:0306)and h ( (cid:126)x ) is the model for the GW signal. We have verified that this choice does not affect our results,since the signals we are studying are short and the effect ofspin-precession is suppressed [41–43]. − . − . − . − .
25 0 .
00 0 .
25 0 .
50 0 .
75 1 . χ eff . . . . . P r o b a b ili t y d e n s i t y uniform χ volumetric FIG. 1. Prior probability density for the effective spin whenemploying a uniform prior on spin magnitudes and directions(black, ‘uniform χ ’), and a uniform prior on spin components(red, ‘volumetric’). In both cases the prior on the componentmasses is flat. The above procedure contains two important ingredi-ents: the prior distribution for the parameters p ( (cid:126)x ) and awaveform model for the GW signal h ( (cid:126)x ). For the prior,we select a uniform distribution for the sky location andthe orientation of the source, a uniform-in-volume distri-bution for the distance, and a uniform distribution for thecomponent masses. We explore two prior distributions forthe spin angular momenta. The first (a ‘uniform χ ’ prior)assumes that the spin magnitude and directions have auniform distribution, p ( χ ) dχ ∝ dχ (this is the defaultchoice for most GW analyses). The second (a ‘volumetric’prior) assumes that the individual spin components areuniformly distributed, p ( χ ) dχ ∝ χ dχ . The resultingprior distribution for the effective spin from these twochoices is plotted in Fig. 1. Both priors favor small effec-tive spins, though the volumetric prior has more supportat high χ eff . These prior choices affect parameter infer-ence [47, 48], and we discuss their impact on measuringlarge spins in Sec. III.We employ two waveform models in the LALInference analysis,
IMRPhenomPv2 [49] and
SEOBNRv4 [38], whichboth include the inspiral, merger and ringdown phases ofa BBH coalescence. Both models have been extensivelyused for the analysis of GW signals, see for exampleRefs. [5, 6].
IMRPhenomPv2 includes the effects of spin-precession in an effective way by parameterizing it througha single effective parameter χ p [50]. SEOBNRv4 , on theother hand, assumes that the spins remain aligned withthe orbital angular momentum throughout the binaryevolution. Both models have been calibrated againstnonprecessing NR simulations (including a simulationwith both spins at 0 .
994 in the case of
SEOBNRv4 ) andhave been shown to match well the predictions of NR [38].Neither model results in systematic biases in the caseof GW150914 [51, 52]. We choose to work with bothwaveform models both for computational convenience andas an independent cross-check of our results.
RecoveredSXS:BBH: Parameter Injected ‘uniform χ ’ ‘volumetric’0305 qM (M (cid:12) ) χ eff χ z χ z . +2 . − . − . +0 . − . − . +0 . − . − . +0 . − . (0.65,1)70 . +2 . − . − . +0 . − . . +0 . − . − . +0 . − . qM (M (cid:12) ) χ eff χ z χ z . +3 . − . . +0 . − . . +0 . − . . +0 . − . (0.63,1)69 . +2 . − . . +0 . − . . +0 . − . − . +1 . − . qM (M (cid:12) ) χ eff χ z χ z . +1 . − . . +0 . − . (0.89,1)(0.84,1) (0.61,1)70 . +1 . − . . +0 . − . (0.94,1)(0.84,1)1137 qM (M (cid:12) ) χ eff χ z χ z . +3 . − . − . +0 . − . (-1,-0.65)(-1,-0.59) (0.77,1)74 . +2 . − . − . +0 . − . (-1,-0.57)(-1,-0.60)TABLE II. Injected and recovered parameters for the four SXSsimulations we study. For each simulated signal (first column)we quote the injected value (third column) and the recoveredvalues (fourth and fifth column) for the mass ratio, the totalmass, the effective spin, and the two spin components along theorbital angular momentum (third column). The fourth columnshows results obtained with IMRPhenomPv2 and the ‘uniform χ ’ prior, while the fifth column presents results with SEOBNRv4 and the ‘volumetric’ prior. The recovered values we quote areeither median and 90% credible intervals or one-sided 90%credible intervals, depending on whether the correspondingposterior rails agains a prior boundary, as further explainedin, for example, [53].
III. RESULTS
In this section, we present the results of the
LALInference parameter estimation study performed onthe simulated signals described in Sec. II, and we discussour ability to robustly characterize nearly extremal BHsin GW observations. Our results indicate that a standardparameter estimation study, such as the one employedby the LIGO and VIRGO collaborations, can lead to areasonable estimation of the total mass, mass ratio, and ef-fective spin of nearly-extremal BHs. However, we recovera systematic offset in χ eff away from extremality, whichis compensated by a systematic bias in the total mass,an outcome of the mass–spin degeneracy. Our parameterestimates are summarized in Table II for both priors ofFig. 1. − . − . − . − .
25 0 .
00 0 .
25 0 .
50 0 .
75 1 . χ eff . . . . . . . P r o b a b ili t y d e n s i t y SXS:BBH:0305SXS:BBH:0306SXS:BBH:1124SXS:BBH:1137
FIG. 2. Marginalized posterior probability density for theeffective spin for simulated signals at an SNR of 25 (solid lines)and 12 (dotted lines), and a total mass of 70 M (cid:12) . The dataare analyzed with IMRPhenomPv2 and the ‘uniform χ ’ prior ofFig. 1 (see Fig. 7 for a reanalysis with the ‘volumetric’ prior).The vertical dashed lines denote the true values of the effectivespin. In all cases, the effective spin is measured, though thismeasurement is biased when the true value of χ eff is close to ±
1. For small values of the true effective spin, the posteriorbecomes more narrow as the SNR of the signal increases. Forlarge (absolute value) effective spins, on the other hand, theposterior both becomes more narrow and shifts towards thetrue value as the signal becomes stronger.
A. Source characterization
The effective spin χ eff is one of the best measured spinparameters with GWs. Therefore, it is commonly em-ployed to characterize spin measurability and to studythe formation channels of BBHs. Figure 2 shows themarginalized posterior probability density for χ eff for foursimulated signals using the NR simulations of Table I,analyzed with the spin-precessing model IMRPhenomPv2 .In all cases, the effective spin posterior is significantlydifferent than the employed ‘uniform χ ’ prior (see Fig. 1)indicating that the posteriors are data driven. However,this measurement is not accurate in the case where thetrue χ eff value is close to the edges of its prior range.Specifically, for cases SXS:BBH:1124 and SXS:BBH:1137,the true value is outside the 99% posterior credible in-terval, consistent also with the findings of Ref. [54]. Inthe next section we discuss this bias and its dependenceon the specific form of the default, ‘uniform χ ’ spin prioremployed here, which disfavors large χ eff values.Regarding the mass parameters, Fig. 3 shows the two-dimensional posterior for the effective spin and the massratio (top panel), and the total mass of the system (bot-tom panel) . It is well known that the effective spin iscorrelated with either the mass ratio or the total mass,depending on the duration of the signal [46]. For longer In this and all similar two-dimensional plots with multiple levelcontours each line corresponds to a 10% increment in the proba-bility. − . − . − .
25 0 .
00 0 .
25 0 .
50 0 .
75 1 . χ eff . . . . . q SXS:BBH:0305 SXS:BBH:0306SXS:BBH:1124SXS:BBH:1137
66 68 70 72 74 76 78 M ( M (cid:12) ) − . . . . χ e ff SXS:BBH:0305SXS:BBH:0306SXS:BBH:1124SXS:BBH:1137
FIG. 3. Marginalized two-dimensional posterior probabilitydensity for the effective spin parameter and the mass ratio (toppanel) and for the effective spin and the total mass (bottompanel) for four simulated signals at an SNR of 25 and a totalmass of 70 M (cid:12) . The data are analyzed with IMRPhenomPv2 andthe ‘uniform χ ’ prior of Fig. 1. The true value is denoted witha cross of the same color as the corresponding contours. Forshort signals such as the ones studied here, the effective spin ispredominantly correlated with the total mass, as demonstratedin the bottom panel. This correlation is almost broken forthe longest duration signal (SXS:BBH:1124, blue posterior)for which the effective spin shows a small correlation with themass ratio (top panel). signals that include a long inspiral phase, the effectivespin is correlated with the mass ratio, as they both af-fect the GW phase at the same post-Newtonian order.On the other hand, if a signal consists primarily of themerger phase, the effective spin is correlated with thetotal mass, since they both affect the frequency of themerger. This trend is visible in Fig. 3, where the M − χ eff correlation is more pronounced than the q − χ eff one for allsignals other than SXS:BBH:1124. Since SXS:BBH:1124has a large positive spin angular momentum it is sub-ject to the effect commonly called “orbital hangup”, anoutcome of post-Newtonian spin-orbit coupling [55, 56](cf. the discussion in Sec. 4.2 of [36] and the referencestherein). This makes SXS:BBH:1124 last longer and bemore inspiral-dominated, and hence more susceptible tothe q– χ eff correlation.Finally, the properties of the final remnant BH are ex-amined in Fig. 4 which shows the marginalized posteriordistribution for the remnant mass and spin. As expected
60 62 64 66 68 70 72 74 76 M f ( M (cid:12) ) . . . . . . . . χ f SXS:BBH:0305SXS:BBH:0306SXS:BBH:1124SXS:BBH:1137
FIG. 4. Marginalized two-dimensional posterior probabilitydensity for the mass and the spin of the remnant BH for foursimulated signals at an SNR of 25 and a total mass of 70 M (cid:12) .The data are analyzed with IMRPhenomPv2 and the ‘uniform χ ’ prior of Fig. 1. The true value is denoted with a cross ofthe same color as the corresponding contours. from the discussion of Fig. 3, reliably extracting the prop-erties of the final BH is challenging if the component spinsare large. Specifically, the true values are within the 90%posterior credible region only in the SXS:BBH:0305 case. B. Spin measurability
In the following we consider the measurability of var-ious spin parameters in more detail. Figure 5 presentsposterior probabilities for the binary spin componentsalong the orbital angular momentum (top panel) and χ eff and χ p (bottom panel). Recall that the spin parameter χ p quantifies the amount of spin-precession present in thesystem [50]. In each panel, we show results for all four sim-ulated signals at SNR 25; the true parameters are shownas crosses in colors matching the corresponding contours.We find that the large individual spin components canonly robustly be measured when both spins are large andparallel to each other. Conversely, configurations withspins antiparallel to each other are recovered as consistentwith slowly-spinning binaries, as also alluded to by Fig. 2.The bottom panel of Fig. 5 shows that the χ p posteriorsextend to large values of χ p . However, comparison ofthese posteriors with the χ p prior shows that the poste-rior is prior-dominated and we cannot constrain χ p fromthe data [57].Figure 6 examines the individual spin magnitudes andshows contours of the two-dimensional posterior probabil-ity density for the individual spin magnitudes for the foursimulated signals with SNR 25. Crosses indicate the truevalues for the spins. The recovered individual spins arehigh when the true spins are nearly extremal and parallelto each other but not when the true spins are antiparallelto each other. This suggests that the individual spinmagnitudes of rapidly spinning BHs can only be reliablymeasured if the spins point in the same direction, creat- − . − . − .
25 0 .
00 0 .
25 0 .
50 0 .
75 1 . χ z − . − . − . − . . . . . . χ z SXS:BBH:0305SXS:BBH:0306 SXS:BBH:1124SXS:BBH:1137 . . . . . χ p − . . . . χ e ff SXS:BBH:0305SXS:BBH:0306SXS:BBH:1124SXS:BBH:1137
FIG. 5. Marginalized two-dimensional posterior probabilitydensity for the binary components’ spins along the orbitalangular momentum (top panel) and for the effective spin χ eff and χ p (bottom panel). The data are analyzed with IMRPhenomPv2 and the uniform prior of Fig. 1. The spincomponents are not recovered accurately, though the bias isless pronounced when the individual spins are large and bothparallel or both antiparallel to the orbital angular momentum. ing a larger effective spin χ eff , in agreement with Fig. 5.The difficulty of measuring individual spin magnitudes ingeneral has been previously discussed in Ref. [58]. C. Effect of spin prior
The accuracy of the measurement of the effective spinparameter in Fig. 2 is poor for the two cases where thetrue value is close to ±
1. In this section, we discuss theeffect of the spin prior on the measurement of large spinvalues [47, 48, 59].Returning to Fig. 2, the dotted lines show the marginal-ized posteriors for χ eff for signals of SNR 12. Strongersignals enable better parameter measurement and morenarrow posterior distributions. This expectation is con-firmed for all four systems studied here. Moreover, in thecase of SXS:BBH:1124 and SXS:BBH:1137 the posteriornot only becomes more narrow, but it also shifts closer tothe true value demonstrating the difficulty of measuringlarge spins. . . . . . . . χ . . . . . . . χ SXS:BBH:0305 . . . . . . χ . . . . . . χ SXS:BBH:0306 .
92 0 .
93 0 .
94 0 .
95 0 .
96 0 .
97 0 .
98 0 .
99 1 . χ . . . . . . . χ SXS:BBH:1124 . . . . . . χ . . . . . . χ SXS:BBH:1137
FIG. 6. Marginalized two-dimensional posterior probability density for the binary components’ spin magnitudes for foursimulated signals with an SNR of 25 and a total mass of 70 M (cid:12) . Here χ and χ are the spins of the larger and smaller BH,respectively. The data is analyzed with IMRPhenomPv2 and the uniform prior of Fig. 1. The true value is denoted with a crosssymbol. Large individual spins can be measured when the spins are either both parallel or both antiparallel to the orbitalangular momentum (1124 and 1137) but not when one spin is parallel and the other antiparallel (0305 and 0306).
To explore the effect of prior we employ the ‘volumetric’prior of Fig. 1, which results in higher prior probabilityat higher spins, as demonstrated in Fig. 1. Figure 7shows the posterior distribution for the effective spin forsignals analyzed with the spin-aligned waveform model
SEOBNRv4 with the ‘uniform χ ’ (solid lines) and the ‘vol-umetric’ (dotted lines) spin prior . In the SXS:BBH:0305and SXS:BBH:0306 cases, all posteriors are very similar,suggesting that the prior distribution has a lesser effecton the posterior when the effective spin is small.In the case where χ eff ∼ ± Despite the ‘uniform χ ’ and ‘volumetric’ priors being derived inthe context of 3-dimensional spin vectors, we can still apply themto spin-aligned waveform models that only include a single spindegree of freedom, the spin component along the orbital angularmomentum χ iz . In that case the prior on the sole spin degreeof freedom is the same as the prior on the χ iz spin componentunder the ‘uniform χ ’ or ‘volumetric’ priors. − . − . − . − .
25 0 .
00 0 .
25 0 .
50 0 .
75 1 . χ eff . . . . . . . . P r o b a b ili t y d e n s i t y SXS:BBH:0305SXS:BBH:0306SXS:BBH:1124SXS:BBH:1137
FIG. 7. Effective spin posteriors for two choices of the spinprior. Solid lines indicate the ‘uniform χ ’ prior and dottedlines denote the ‘volumetric’ prior. This analysis is performedwith the spin-sligned waveform model SEOBNRv4 . shows contours for the two-dimensional posterior proba-bility density for the spin components along the orbitalangular momentum. As expected, all posteriors derivedwith the ‘volumetric’ prior have more support for largevalues of the spin components. − . − . − . − .
25 0 .
00 0 .
25 0 .
50 0 .
75 1 . χ z − . − . − . − . . . . . . χ z SXS:BBH:0305 SXS:BBH:0306SXS:BBH:1124SXS:BBH:1137
FIG. 8. Impact of spin prior on χ z − χ z recovery withthe aligned-spin waveform model SEOBNRv4 . Shown are incre-mental contours for the ‘uniform χ ’ prior and a solid-line 90%credible level contour for the ‘volumetric’ prior. D. Effect of signal duration
Due to the finite length of the NR data, all results pre-sented in the above subsections assumed a total massof 70 M (cid:12) , which is comparable to the total mass ofGW150914 [3]. If the total mass of the system is lowerthan this value, the start of the numerical waveform fallswithin the sensitive frequencies of the detector, poten-tially affecting the results of parameter estimation [60].To study the effect of the signal duration on our results,instead, we use the waveform model IMRPhenomPv2 tosimulate the GW data with parameters equal to thoseof SXS:BBH:1124 and SXS:BBH:1137 but with a totalmass of 30 , , and 70 M (cid:12) . We employ the same modelfor signal recovery and find that the resulting posteriorsare very similar, suggesting that our main conclusions areunaffected by the signal duration. E. Model accuracy
Our study suggests that current analyses are sub-optimal for characterizing signals with large spins. How-ever, the waveform models used for these analyses mayalso lose accuracy at this challenging region of the param-eter space [38]. This prompts the question: is it hard tomeasure large spins because of the posterior properties orbecause the waveform models employed misbehave? Inorder to fully address this question we would have to per-form parameter estimation directly using NR waveforms,something that is currently impossible for the region ofthe parameter space we are interested in. However, belowwe discuss evidence suggesting that the difficulty to mea-sure large spins has less to do with the accuracy of themodels, and more with the properties of the likelihoodfunction and the prior choices.First, when the SNR of the signal is increased, theposterior distribution for χ eff in the SXS:BBH:1137 andSXS:BBH:1124 cases shifts towards the true value, as shown in Fig. 2. Since systematic errors caused by modelinaccuracies do not depend on the SNR, the shift in theposterior suggests it is mainly the prior that keeps theposteriors away from large χ eff values.Second, we repeat the analysis described above andcompute the posterior for the effective spin parameterusing a simulated signal created with the numerical wave-forms and with the IMRPhenomPv2 waveform model. Wefind a large similarity between the posterior obtainedwith the different data, as shown in Fig. 9. Specifically,the shift in the posterior due to the change of data inFig. 9 is smaller than the shift due to changing the spinprior in Fig. 7. This suggests that at the injected param-eter values the NR waveform and the data created with
IMRPhenomPv2 do not possess noticeable differences as faras parameter estimation is concerned. − . − . − . − .
25 0 .
00 0 .
25 0 .
50 0 .
75 1 . χ eff . . . . . . . . P r o b a b ili t y d e n s i t y SXS:BBH:1124, SXS injectionSXS:BBH:1124, IMR injectionSXS:BBH:1137, SXS injectionSXS:BBH:1137, IMR injection
FIG. 9. Similar to Fig. 2 for signals created with NR data(solid lines), and with the
IMRPhenomPv2 waveform model(dotted lines). The similarity between the solid and the dottedcurves suggest that systematic difference between NR andwaveform models are not the dominant cause of our conclusionthat large spins are difficult to measure.
Third, we employ a figure of merit commonly used inwaveform modeling, namely the overlap between the signaland the template, defined as ( d | h ( (cid:126)x )) / (cid:112) ( d | d )( h ( (cid:126)x ) | h ( (cid:126)x )).Figure 10 shows a scatter plot of the posterior samples forthe SXS:BBH:1137 case of the lower panel of Fig. 3. Thesamples are colored by their overlap value; we find overlapsaround 99 .
5% in the region of the injected parameters,and they drop as we move away from the true parameters.We obtain similar results for the other three NR signalsstudied here and the
SEOBNRv4 model. The high valueof overlap further suggests that systematic biases aresubdominant for this region of the parameter space andfor this SNR value [61].
IV. CONCLUSIONS
In this paper, we assess the prospects of extractingthe spins of nearly extremal BHs in binaries with GWmeasurements. We find that measurement of large spins ischallenging. Favorable conditions occur when both spinsare large and parallel to each other, but even in this case
70 75 80 85 M ( M (cid:12) ) − . − . − . − . − . χ e ff . . . . . . . FIG. 10. Similar to the bottom panel of Fig. 3 for the case ofSXS:BBH:1137 with 5000 scattered posterior samples coloredby the value of their overlap with the simulated data. Theoverlap achieved close to the injected value is in the 99.5%range. our posteriors are biased away from extremal effectivespins. We argue that this is due to the commonly usedspin priors that disfavor large spins.Additionally, extremal spins are close to the edge ofthe spin priors. This situation is similar to the case ofmeasuring the mass ratio (or the symmetric mass ratio)of equal-mass systems. In fact, when the posterior dis-tribution of a parameter rails agains a prior edge, it iscustomary to use one-sided credible intervals or highest-probability-density intervals (for an extended discussion,see [53]). However, we find this is not the case for the effective spin, since its posterior typically does not railagainst the prior edge (see Fig. 2). We attribute this tothe spin prior, which drops to vanishingly small values as χ eff → ±
1. In order to overcome this trend and obtaina likelihood-dominated effective spin posterior, a signalwith large SNR is needed.Our results showcase again the importance of priorsand prior bounds in GW inference and suggest the use ofa wide range of spin priors. This will not only allow usto study physical effects such as the large spins describedhere, but can also enable further studies such as thehierarchical analysis described in [20].
V. ACKNOWLEDGMENTS
We are pleased to thank Sebastian Khan and JacobLange for useful discussions on producing simulated GWsignals with NR data. We would also like to thank JoshuaSmith and Jocelyn Read for helpful discussions and LeoStein and Juan Calderon Bustillo for comments on themanuscript. This work was supported in part by Na-tional Science Foundation grants PHY-1606522 and PHY-1654359 to Cal State Fullerton. We gratefully acknowl-edge support for this research at Caltech from NSF grantsPHY-1404569, PHY-1708212, and PHY-1708213 and theSherman Fairchild Foundation and at Cornell from NSFGrant PHY-1606654 and the Sherman Fairchild Founda-tion. [1] J. Aasi et al. (LIGO Scientific), Class. Quant. Grav. ,074001 (2015), arXiv:1411.4547 [gr-qc].[2] F. Acernese et al. (VIRGO), Class. Quant. Grav. ,024001 (2015), arXiv:1408.3978 [gr-qc].[3] B. P. Abbott et al. (LIGO and Virgo Scientific Col-laboration), Phys. Rev. Lett. , 061102 (2016),arXiv:1602.03837 [gr-qc].[4] B. P. Abbott et al. (LIGO and Virgo Scientific Col-laboration), Phys. Rev. Lett. , 241103 (2016),arXiv:1606.04855 [gr-qc].[5] B. P. Abbott et al. (LIGO and Virgo Scientific Col-laboration), Phys. Rev. Lett. , 221101 (2017),arXiv:1706.01812 [gr-qc].[6] B. P. Abbott et al. (Virgo, LIGO Scientific), Phys. Rev.Lett. , 141101 (2017), arXiv:1709.09660 [gr-qc].[7] B. P. Abbott et al. (Virgo, LIGO Scientific), Astrophys.J. , L35 (2017), arXiv:1711.05578 [astro-ph.HE].[8] J. Abadie, B. P. Abbott, R. Abbott, M. Abernathy, T. Ac-cadia, F. Acernese, C. Adams, R. Adhikari, P. Ajith,B. Allen, and et al., Classical and Quantum Gravity ,173001 (2010), arXiv:1003.2480 [astro-ph.HE].[9] K. Belczynski, D. E. Holz, T. Bulik, andR. O’Shaughnessy, Nature , 512 (2016),arXiv:1602.04531 [astro-ph.HE]. [10] C. L. Rodriguez, C.-J. Haster, S. Chatterjee, V. Kalogera,and F. A. Rasio, Astrophys. J. , L8 (2016),arXiv:1604.04254 [astro-ph.HE].[11] S. Stevenson, A. Vigna-Gmez, I. Mandel, J. W. Barrett,C. J. Neijssel, D. Perkins, and S. E. de Mink, (2017),10.1038/ncomms14906, [Nature Commun.8,14906(2017)],arXiv:1704.01352 [astro-ph.HE].[12] C. L. Rodriguez, M. Zevin, C. Pankow, V. Kalogera,and F. A. Rasio, Astrophys. J. , L2 (2016),arXiv:1609.05916 [astro-ph.HE].[13] D. Kushnir, M. Zaldarriaga, J. A. Kollmeier, and R. Wald-man, Mon. Not. Roy. Astron. Soc. , 844 (2016),arXiv:1605.03839 [astro-ph.HE].[14] D. Gerosa, M. Kesden, E. Berti, R. O’Shaughnessy,and U. Sperhake, Phys. Rev. D87 , 104028 (2013),arXiv:1302.4442 [gr-qc].[15] D. Gerosa and E. Berti, Phys. Rev.
D95 , 124046 (2017),arXiv:1703.06223 [gr-qc].[16] M. Fishbach, D. E. Holz, and B. Farr, Astrophys. J. ,L24 (2017), arXiv:1703.06869 [astro-ph.HE].[17] S. Vitale, R. Lynch, R. Sturani, and P. Graff, Class.Quant. Grav. , 03LT01 (2017), arXiv:1503.04307 [gr-qc].[18] S. Stevenson, C. P. L. Berry, and I. Mandel, Mon. Not.Roy. Astron. Soc. , 2801 (2017), arXiv:1703.06873 [astro-ph.HE].[19] C. Talbot and E. Thrane, Phys. Rev. D96 , 023012 (2017),arXiv:1704.08370 [astro-ph.HE].[20] W. M. Farr, S. Stevenson, M. Coleman Miller, I. Man-del, B. Farr, and A. Vecchio, Nature , 426 (2017),arXiv:1706.01385 [astro-ph.HE].[21] B. Farr, D. E. Holz, and W. M. Farr, (2017),arXiv:1709.07896 [astro-ph.HE].[22] I. Mandel and R. O’Shaughnessy,
Numerical relativity anddata analysis. Proceedings, 3rd Annual Meeting, NRDA2009, Potsdam, Germany, July 6-9, 2009 , Class. Quant.Grav. , 114007 (2010), arXiv:0912.1074 [astro-ph.HE].[23] K. Belczynski et al. , (2017), arXiv:1706.07053 [astro-ph.HE].[24] J. Fuller, M. Cantiello, D. Lecoanet, and E. Quataert,Astrophys. J. , 101 (2015), arXiv:1502.07779 [astro-ph.SR].[25] M. C. Miller and J. M. Miller, Phys. Rept. , 1 (2014),arXiv:1408.4145 [astro-ph.HE].[26] L. Gou, J. E. McClintock, R. A. Remillard, J. F. Steiner,M. J. Reid, et al. , (2013), arXiv:1308.4760 [astro-ph.HE].[27] J. E. McClintock, R. Shafee, R. Narayan, R. A. Remil-lard, S. W. Davis, et al. , Astrophys.J. , 518 (2006),arXiv:astro-ph/0606076 [astro-ph].[28] K. Belczynski, T. Bulik, and C. L. Fryer, ArXiv e-prints(2012), arXiv:1208.2422 [astro-ph.HE].[29] E. Racine, Phys. Rev. D78 , 044021 (2008),arXiv:0803.1820 [gr-qc].[30] D. Gerosa, M. Kesden, U. Sperhake, E. Berti, andR. O’Shaughnessy, Phys. Rev.
D92 , 064016 (2015),arXiv:1506.03492 [gr-qc].[31] M. Zaldarriaga, D. Kushnir, and J. A. Kollmeier, Mon.Not. Roy. Astron. Soc. , 4174 (2018), arXiv:1702.00885[astro-ph.HE].[32] K. Hotokezaka and T. Piran, Astrophys. J. , 111(2017), arXiv:1702.03952 [astro-ph.HE].[33] S. eXtreme Spacetimes (SXS) Collaboration, http://black-holes.org/SpEC.html .[34] A. H. Mroue, M. A. Scheel, B. Szil´agyi, H. P. Pfeiffer,M. Boyle, D. A. Hemberger, L. E. Kidder, G. Lovelace,S. Ossokine, N. W. Taylor, A. Zenginoglu, L. T. Buchman,T. Chu, E. Foley, M. Giesler, R. Owen, and S. A. Teukol-sky, Phys. Rev. Lett. , 241104 (2013), arXiv:1304.6077[gr-qc].[35] .[36] M. A. Scheel, M. Giesler, D. A. Hemberger, G. Lovelace,K. Kuper, M. Boyle, B. Szil´agyi, and L. E. Kidder, Class.Quant. Grav. , 105009 (2015), arXiv:1412.1803 [gr-qc].[37] G. Lovelace et al. , Class. Quant. Grav. , 244002 (2016),arXiv:1607.05377 [gr-qc].[38] A. Bohe et al. , Phys. Rev. D95 , 044028 (2017),arXiv:1611.03703 [gr-qc]. [39] B. P. Abbott et al. (VIRGO, LIGO Scientific),(2013), 10.1007/lrr-2016-1, [Living Rev. Rel.19,1(2016)],arXiv:1304.0670 [gr-qc].[40] S. Nissanke, D. E. Holz, S. A. Hughes, N. Dalal, and J. L.Sievers, Astrophys. J. , 496 (2010), arXiv:0904.1017[astro-ph.CO].[41] K. Chatziioannou, N. Cornish, A. Klein, and N. Yunes,Phys. Rev.
D89 , 104023 (2014), arXiv:1404.3180 [gr-qc].[42] K. Chatziioannou, N. Cornish, A. Klein, and N. Yunes,Astrophys. J. , L17 (2015), arXiv:1402.3581 [gr-qc].[43] B. Farr et al. , Astrophys. J. , 116 (2016),arXiv:1508.05336 [astro-ph.HE].[44] P. Schmidt, I. W. Harry, and H. P. Pfeiffer, (2017),arXiv:1703.01076 [gr-qc].[45] J. Veitch et al. , Phys. Rev.
D91 , 042003 (2015),arXiv:1409.7215 [gr-qc].[46] C. Cutler and E. E. Flanagan, Phys. Rev. D , 2658(1994).[47] S. Vitale, D. Gerosa, C.-J. Haster, K. Chatziioannou,and A. Zimmerman, Phys. Rev. Lett. , 251103 (2017),arXiv:1707.04637 [gr-qc].[48] D. Gerosa, S. Vitale, C.-J. Haster, K. Chatziioannou,and A. Zimmerman, in IAU Symposium 338: Gravi-tational Wave Astrophysics: Early Results from GWSearches and Electromagnetic Counterparts Baton Rouge,LA, USA, October 16-19, 2017 (2017) arXiv:1712.06635[astro-ph.HE].[49] M. Hannam, P. Schmidt, A. Boh, L. Haegel, S. Husa,F. Ohme, G. Pratten, and M. Prrer, Phys. Rev. Lett. , 151101 (2014), arXiv:1308.3271 [gr-qc].[50] P. Schmidt, F. Ohme, and M. Hannam, Phys. Rev.
D91 ,024043 (2015), arXiv:1408.1810 [gr-qc].[51] B. P. Abbott et al. (Virgo, LIGO Scientific), Class. Quant.Grav. , 104002 (2017), arXiv:1611.07531 [gr-qc].[52] J. Caldern Bustillo, P. Laguna, and D. Shoemaker, Phys.Rev. D95 , 104038 (2017), arXiv:1612.02340 [gr-qc].[53] B. P. Abbott et al. (Virgo, LIGO Scientific), (2018),arXiv:1805.11579 [gr-qc].[54] C. Afle et al. , (2018), arXiv:1803.07695 [gr-qc].[55] T. Damour, Phys. Rev. D , 124013 (2001), arXiv:gr-qc/0103018 [gr-qc].[56] L. E. Kidder, Phys. Rev. D52 , 821 (1995), arXiv:gr-qc/9506022.[57] B. P. Abbott et al. (Virgo, LIGO Scientific), Phys. Rev.Lett. , 241102 (2016), arXiv:1602.03840 [gr-qc].[58] M. Purrer, M. Hannam, and F. Ohme, Phys. Rev.
D93 ,084042 (2016), arXiv:1512.04955 [gr-qc].[59] A. R. Williamson, J. Lange, R. O’Shaughnessy, J. A.Clark, P. Kumar, J. Caldern Bustillo, and J. Veitch,Phys. Rev.