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Supplement: a New Spin on LIGO-Virgo Binary Black Holes

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Supplement: a New Spin on LIGO-Virgo Binary Black Holes Supplement: A new spin on LIGO-Virgo binary black holes Sylvia Biscoveanu,1, 2, ∗ Maximiliano Isi,1, 2, y Salvatore Vitale,1, 2 and Vijay Varma3, 4, z 1LIGO Laboratory, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 2Department of Physics and Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge, MA 02139, USA 3TAPIR, California Institute of Technology, Pasadena, CA 91125, USA 4Department of Physics, and Cornell Center for Astrophysics and Planetary Science, Cornell University, Ithaca, New York 14853, USA (Dated: March 18, 2021) Parameter estimation methods We choose to sample in µ(χ) and σ2(χ), imposing con- straints such that α; β > 1 to restrict the parameter space To analyze the simulated signals, we perform Bayesian to only nonsingular Beta distributions. parameter estimation using the standard Gaussian likeli- The distributions for χA and χB obtained by applying hood for gravitational-wave data [1,2]: order statistics to the Beta distribution for χ1=2 are given by: 2 Y 2 2 di(fj) h(fj; θ) (di θ) = exp j − j ; p(χA) = 2 p(χ1=2 α; β) CDF(χ1=2 α; β) ; (5) L j πT Sn(fj) − TSn(fj) j j j p(χB) = 2 p(χ α; β) 1 CDF(χ α; β) ; (6) (1) 1=2j − 1=2j assuming χA is the maximum and χB is the minimum of where T is the duration of the data segment being ana- two draws from p(χ1=2 α; β). CDF(χ1=2) is the cumula- lyzed, Sn(fj) is the noise power spectral density of the j tive distribution function for χ1=2 given by the regularized detector, di(fj) is the strain data for event i, and h(fj; θ) incomplete Beta function with parameters (α; β). is the waveform model for the compact binary source. We We further assume that the primary mass distribu- simulate the measurement using the LALInference algo- tion is described by the sum of a truncated power-law rithm [2] and the numerical relativity surrogate waveform with low-mass smoothing and a Gaussian component [6]. model NRSur7dq4 [3]. In order to obtain posterior sam- The hyperparameters describing this model are the slope ples for the parameters θ using the likelihood in Eq.1, we αm, upper and lower cutoffs mmax and mmin, the low- impose priors that are uniform in the component masses mass smoothing parameter δm, the peak and width of m ; m between 10 M and 240 M , with constraints 1 2 the Gaussian component µm and σm, and the mixing on the total mass between 70 M and 240 M and on fraction between the two components λpeak. The mass mass ratio q between 0.2 and 1. The luminosity distance ratio distribution is modeled as a power law with slope βq. prior is d2 over the range 1{7000 Mpc. / L This corresponds to Model C from [5], dubbed\power-law For the spin tilt population inference, we simultaneously + peak". We use this same mass model when computing fit the mass and spin magnitude distributions. We follow the spin magnitude distribution using prior samples for [4, 5] and assume that both of the black hole (BH) spin the GWTC-2 events, shown in the dashed black line in magnitudes under the mass sorting are drawn from the Fig. 3 in the main text. same Beta distribution with hyperparameters α and β, Following [7], the distribution for spin tilt angles is given α−1 β−1 by the sum of an isotropic component and a preferentially χ1=2 (1 χ1=2) aligned component, which is composed of the product of p(χ1=2 α; β) = − ; (2) j B(α; β) two truncated Gaussians peaked at cos ti = 1 for each tilt angle: where B(α; β) is the Beta function. We restrict the priors on the Beta function parameters to exclude values of 1 ξ p(cos t1=A; cos t2=B σ1=A; σ2=B; ξ) = − + (7) α; β 1 corresponding to singular Beta distributions. j 4 ≤ 2 2 This means that p(χ) must peak within 0 < χ < 1, as 2ξ Y exp( (1 cos ti) =(2σi )) − − ; nonsingular Beta distributions vanish at those values. π p i2f1=A;2=Bg σierf( 2/σi) The Beta distribution described in Eq.2 can also be parameterized in terms of its mean and variance: where the hyperparameter ξ gives the mixture fraction between the two components. The complete population α µ(χ) = ; (3) model, π(θ Λ), is given by the product of Eqs.2,7, and α + β the \power-lawj + peak" mass distribution. 2 αβ Using the population model π(θ Λ) to describe the dis- σ (χ) = 2 : (4) (α + β) (α + β + 1) tribution of individual-event parametersj θ, the likelihood 2 of observing the hierarchical parameters Λ for a data set d consisting of Ndet detected events is given by: f g Ndet R Y (di θ)π(θ Λ) ( d Λ) L j j (8) L f gj / α(Λ) i=1 where α(Λ) represents the detectable fraction of events prior samples assuming the individual-event parameters are drawn from spin sorting (A, B) distributions specified by hyperparameters Λ [8{13]. We mass sorting (1, 2) use the sensitive spacetime volume estimates released by the LVC in [14], which for the GWTC-2 analysis were .2 determined through injection campaign [15] and for the 3 .4 GWTC-1 events were obtained using simulated data. We /B 2 2 calculate α(Λ) using the formalism described in [16]. We σ .6 do not account for the selection biases due to the spin 1 8 parameters, since those have a much smaller effect than 0. the mass parameters [17]. 8 6 4 2 8 6 4 2 The likelihood in Eq.8 is evaluated using a Monte Carlo 0. 1. 2. 3. 0. 1. 2. 3. integral over the individual-event parameter posteriors released by the LVC for the binary BH (BBH) events σ1/A σ2/B included in GWTC-1 [18, 19] and GWTC-2 [15, 20]. For the GWTC-1 events, we use the samples obtained with FIG. 1. Corner plot comparing the inference on the spin tilt the IMRPhenomPv2 waveform model, while for GWTC-2 hyperparameters using the mass sorted tilts θ1 and θ2 and spin we use the \Publication" posterior samples presented in sorted tilts θA and θB for the GWTC-1 events. The posteriors [15], which in most cases use a combination of waveform obtained using prior samples for the individual events are models including the effects of spin precession and higher- shown in grey. order multipoles. For GWTC-2, we only analyze the 44 confident BBH detections with a false alarm rate < 1 yr−1 and exclude the events where at least one of the compact is excluded at > 90% credibility for both the mass and objects has considerable posterior support below 3 M . spin sortings. The posterior for σB is less constrained We use the dynesty [21] sampler, as implemented in the than that for σ2|indicating that for this analysis, the GWPopulation [22] package, to obtain hyperparameter information on the tilt angles at the population level is posterior samples. predominantly obtained from the measurement of the The posterior population distribution calculated using highest spinning object, rather than the most massive. A the hyperparameter posteriors obtained using the likeli- similar trend is present in the σ posteriors for the full hood in Eq.8 is given by: GWTC-2 analysis shown in Fig. 4 in the main text. Z PPD(θ d ) = π(θ Λ)p(Λ d )dΛ: (9) jf g j jf g Additional results for individual sources The priors and posterior results for all hyperparameters except σ1=A and σ2=B for the spin-sorted tilt inference For certain sources, we note significant differences in the performed using the GWTC-2 BBH events are shown in posteriors obtained with the two different waveform mod- TableI. The priors are uniform for all parameters and els applied to BBHs in GWTC-1: IMRPhenomPv2 [23{ identical for both the mass and spin-sorted tilt analysis. 25], which uses an effective precessing spin model, and We use uniform priors over the range (0; 4) for the tilt SEOBNRv3, which uses a fully precessing spin model [26{ σ parameters. The posterior for the spin tilt mixing 28]. For GW150914, the one-dimensional posterior for χA fraction ξ peaks at the upper edge of its prior, indicating is much more tightly constrained for SEOBNRv3, with +0:44 +0:45 that a purely isotropic distribution of tilts is statistically χA = 0:39−0:24 compared to χA = 0:49−0:38 for IMRPhe- disfavored by the data. We obtain posteriors for the mass, nomPv2, a feature which is not as easily recognizable in spin magnitude, and ξ hyperparameters comparable to the χ1 posterior. A comparison of the spin magnitudes those quoted in [17]. The results for these parameters obtained using both waveform models is shown in Fig.2. change negligibly under the mass and spin sortings. Similarly for GW170814, the posterior for χA turns over The corner plot of the tilt σ hyperparameters for the at around χA 0:5 for SEOBNRv3, but not for IMRPhe- ∼ GWTC-1 analysis is shown in Fig.1. Even with only nomPv2 (Fig.3). The two-dimensional χA; χB posterior the GWTC-1 events, the corner of parameter space at recovered with SEOBNRv3 is much more tightly clustered σ1=A = σ2=B = 0 representing a fully-aligned population around low spins for GW170814 and also for GW170818, 3 Parameter Prior GWTC-1 GWTC-2 +4:87 +0:77 αm U(-4, 12) 6:41−4:44 2:96−0:63 +5:65 +2:26 βm U(-4, 12) 5:60−5:73 1:01−1:41 +35:92 +12:10 mmax U(30 M ; 100 M ) 59:80−26:66 M 86:30−12:66 M +1:35 +1:38 mmin U(2 M ; 10 M ) 7:40−3:31 M 4:71−1:84 M +5:28 +4:24:35 δm U(0 M ; 10 M ) 2:58−2:38 M 4:59−3:98 M +5:71 +3:52 µm U(20 M ; 50 M ) 28:60−6:98 M 32:42−5:91 M +3:37 +4:21 σm U(0:4 M ; 10 M ) 6:13−4:18 M 5:17−3:85 M +0:40 +0:13 λpeak U(0; 1) 0:19−0:16 0:07−0:05 +0:20 +0:11 µ(χ) U(0; 1) 0:30−0:15 0:31−0:09 2 +0:03 +0:02 σ (χ) U(0; 0:25) 0:02−0:02 0:03−0:02 +0:41 +0:19 ξ U(0; 1) 0:54−0:47 0:79−0:43 TABLE I.
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