Population Properties of Compact Objects from the Second LIGO–Virgo Gravitational-Wave Transient Catalog

• 1.  
  Truncated (four parameters; Appendix B.1). Our simplest mass model, the distribution of primary masses is a power law with hard cutoffs at both low (m_min) and high (m_max) masses. The high-mass cutoff corresponds to the lower edge of the theorized pair-instability gap in the BH mass spectrum (Heger & Woosley 2002; Heger et al. 2003; Woosley & Heger 2015). The mass ratio distribution is also assumed to be a power law (Kovetz et al. 2017; Fishbach & Holz 2017). In Abbott et al. (2019a), it is referred to as "Model B." This model struggles to accommodate high-mass events like GW190521, necessitating more complicated models.
• 2.  
  power law + peak (eight parameters; Appendix B.2). Similar to the Truncated model but with two modifications. At low masses, we implement a smoothing function to avoid a hard cutoff. At high masses, we include a Gaussian peak, initially introduced to model a pileup from pulsational pair-instability supernovae (PPSN; Talbot & Thrane 2018). This model is better able to accommodate the high-mass events that pose a challenge for the Truncated model. In Abbott et al. (2019a), it is referred to as "Model C."
• 3.  
  Broken power law (seven parameters; Appendix B.3). The same as Truncated except, instead of a single power law between m_min and m_max, the model allows for a break in the power law at some mass m_break. This model includes the low-mass smoothing function used in the power law + peak model. The high-mass feature m_break may correspond to the onset of pair-instability, and the second power law can be thought of as either a gradual tapering off or a subpopulation of BHs within the pair-instability mass gap. This model accommodates the high-mass events that pose a challenge for the Truncated model while providing an alternative to the power law + peak model.
• 4.  
  Multipeak (11 parameters; Appendix B.4). This phenomenological model is similar to power law + peak, except we allow for two peaks. The resulting mass distribution can accommodate hierarchical BBH mergers (Miller & Hamilton 2002b; Gültekin et al. 2004; Antonini & Rasio 2016; Rodriguez et al. 2019), in which second-generation mergers create a second high-mass peak in the BH mass spectrum. We use this model to test if GWTC-2 exhibits evidence for hierarchical mergers.

• 1.  
  Default (four parameters; Appendix D.1). This parameterization for the component BH spin magnitudes and tilts was previously used to explore the spin distribution of compact binaries in GWTC-1 (Abbott et al. 2019a). The spin of each component BH in a binary is assumed to be independently drawn from the same underlying distribution. The dimensionless spin magnitude is described using a beta distribution as in Wysocki et al. (2019a). The spin-tilt distribution from Talbot & Thrane (2017) is a mixture model comprising two components: an isotropic component designed to model dynamically assembled binaries and a component in which the spins are preferentially aligned with the orbital angular momentum, as expected for isolated field binaries. The fraction of binaries in the purely aligned subpopulation is denoted as ζ. ²¹⁷ For this latter component, the spin-tilt angles are distributed as a Truncated Gaussian with width ${\sigma }_{t}$ that peaks when the BH spin is aligned to the orbital angular momentum. We use this model in concert with the mass models described above.
• 2.  
  Gaussian (five parameters; Appendix D.2). While the default spin model is physically inspired, this model, based on that of Miller et al. (2020), allows us to fit the distribution of phenomenological spin parameters ${\chi }_{\mathrm{eff}}$ (the effective inspiral spin parameter; Equation (5)) and ${\chi }_{{\rm{p}}}$ (the effective precession spin parameter; Equation (6)), assuming that their distribution is jointly described as a bivariate Gaussian. The ensemble properties of ${\chi }_{\mathrm{eff}}$ and ${\chi }_{{\rm{p}}}$ allow us to conclude that the BBHs in GWTC-2 exhibit general relativistic spin-induced precession of the orbital plane (${\chi }_{{\rm{p}}}\gt 0$), and that some systems have component spins misaligned by more than 90° (${\chi }_{\mathrm{eff}}\lt 0$) relative to the orbital angular momentum.
• 3.  
  Multispin (12 spin parameters, 10 mass parameters; Appendix D.3). This model allows for multiple subpopulations of BBH systems with distinct mass and spin distributions. Specifically, this model assumes a Truncated power-law mass distribution with the additional presence of a 2D Gaussian subpopulation in m₁ and m₂, Truncated such that ${m}_{1}\geqslant {m}_{2}$. While similar to the power law + peak mass model, there is no smooth turn-on, and the mass ratio distribution is allowed to differ between each subpopulation. Most importantly, the two subpopulations have independently parameterized default spin distributions. We use this model to test whether the BBH spin distribution varies as a function of mass, as expected if higher-mass systems are the products of hierarchical mergers.

• 1.  
  Nonevolving (zero parameters). Our default model posits that the merger rate is uniform in comoving volume.
• 2.  
  Power-law evolution (one parameter; Appendix E). Following Fishbach et al. (2018), the merger rate density is described by a power law in $(1+z)$, where z is redshift. Given the finite range of Advanced LIGO and Advanced Virgo to BBH mergers, we only expect to constrain the redshift evolution at redshifts $z\lesssim 1$ (Abbott et al. 2013). The farthest event in our analysis is likely GW190706_222641, at redshift $z={0.71}_{-0.27}^{+0.32}$.

In this subsection, we report results obtained using the mass models described in Section 3.1 (see Figure 1). ²¹⁸ We employ the default spin model and the nonevolving redshift model. The results shown here are marginalized over the hyperparameters of the spin distribution.

A Truncated power law fails to fit the high-mass BBH events. The Truncated model applied in Abbott et al. (2019a) measured the sharp high-mass cutoff to be ${m}_{\max }\,={40.8}_{-4.4}^{+11.8}\ {M}_{\odot }$. When we fit this model to GWTC-2 data, we obtain ${m}_{\max }={78.5}_{-9.4}^{+14.1}\ {M}_{\odot }$, in significant tension (>99% credibility) with the GWTC-1 result; see Figure 2. The Truncated model struggles to accommodate the high-mass systems of GWTC-2. At 99% credibility, three events of GWTC-2 have ${m}_{1}\gt 45\ {M}_{\odot }$ (regardless of whether we use a population-informed prior; Fishbach et al. 2020b). This tension remains (at the >93% credibility level) even when we exclude the highest-mass event, GW190521 (Abbott et al. 2020e, 2020f). The poor fit of the Truncated model is further seen in the posterior predictive check of Figure 19 in Appendix C, which shows that the Truncated model fails to capture the relative excess of observations with ${m}_{1}\sim 30\ {M}_{\odot }$ compared to the number of events with ${m}_{1}\gtrsim 45\ {M}_{\odot }$. Our fit to the Truncated model overpredicts the number of observations with ${m}_{1}\gt 45\ {M}_{\odot }$ relative to the number of observations with $30{M}_{\odot }\lt {m}_{1}\lt 45\ {M}_{\odot }$ (98% credibility).

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The power law + peak, broken power-law, and multipeak models provide better fits to the shape of the mass distribution, particularly at high masses. Although our updated fit to the mass distribution extends to higher masses than the GWTC-1 fit, we find that ${97.1}_{-3.5}^{+1.7} \%$ of BBH systems have primary masses below $45\ {M}_{\odot }$ (power law + peak model), consistent with the GWTC-1 estimate that 99% of primary masses lie below ∼$45\ {M}_{\odot }$ (Abbott et al. 2019a; Kimball et al. 2020a). In Table 2, we provide log Bayes factors (${\mathrm{log}}_{10}{ \mathcal B }$) comparing the mass models. Each Bayes factor is measured relative to the model with the highest Bayesian evidence: power law + peak. In each case, we use the default spin model. For context, ${\mathrm{log}}_{10}{ \mathcal B }\gt 1.5$ is often interpreted as a strong preference for one model over another and ${\mathrm{log}}_{10}{ \mathcal B }\gt 2$ as decisive evidence (Jeffreys 1961).

Table 2. Bayes Factors for Each Mass Model Relative to the Favored power law + peak Model, Which Gives the Highest Bayesian Evidence for GWTC-2

Mass Model	${ \mathcal B }$	${\mathrm{log}}_{10}{ \mathcal B }$
power law + peak	$1.0$	$0.0$
Multipeak	$0.5$	$-0.3$
Broken power law	$0.12$	$-0.92$
Truncated	$0.01$	$-1.91$
power law + peak (${\delta }_{m}=0$)	$0.87$	$-0.06$
Broken power law + peak	$0.74$	$-0.13$
Broken power law (${\delta }_{m}=0$)	$0.35$	$-0.46$
power law + peak (${\lambda }_{\mathrm{peak}}=0$)	$0.05$	$-1.34$

Note. For models that have a smooth turn-on at low masses parameterized by ${\delta }_{m}$, we also compare the corresponding submodel with a sharp minimum mass cutoff (${\delta }_{m}=0$). For the power law + peak model, which includes a fraction ${\lambda }_{\mathrm{peak}}$ of systems in the Gaussian component, we compare the submodel with ${\lambda }_{\mathrm{peak}}=0$. We exclude GW190814 from this analysis.

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While Bayes factors depend on the choice of hyperparameter priors, it is nonetheless possible to see that the Truncated model is disfavored compared to the more complicated models. This inference is driven in part by the fact that, 94% of the time in our posterior predictive checks, the Truncated model overpredicts the number of detections with ${m}_{1}\gt 50\ {M}_{\odot }$. See Appendix C for information about the posterior predictive checks. Meanwhile, there is no strong preference for power law + peak over broken power law or multipeak. We currently lack the resolving power to determine whether the deviations from the Truncated model are best modeled as a break, a Gaussian peak, or two Gaussian peaks. As a further check, we carried out a follow-up analysis using a hybrid broken power law + peak model, which indicated only modest support for a peak on top of the broken power-law distribution (${\mathrm{log}}_{10}{ \mathcal B }=0.79$).

There are features in the BH mass spectrum beyond a power law. Figure 3 shows the astrophysical merger rate density as a function of primary BH mass for the Truncated, power law + peak, multipeak and broken power-law models. Figure 4, meanwhile, shows the posterior predictive distribution for primary masses, including selection effects. Corner plots showing the constraints for the parameters in each model are available in Appendix B; see Figures 16–18. In Appendix C, we show posterior predictive checks for each model (Figures 19 and 23), comparing mock observations predicted by the model to the empirical distribution inferred from GWTC-2.

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We turn first to the broken power-law model (second panel in Figure 3), which is characterized by two spectral indices, ${\alpha }_{1}$ and ${\alpha }_{2}$, with $p({m}_{1})\propto {m}_{1}^{-{\alpha }_{1}}$ for ${m}_{1}\lt {m}_{\mathrm{break}}$ and $p({m}_{1})\propto {m}_{1}^{-{\alpha }_{2}}$ above the break. We find that the data prefer a break at ${m}_{\mathrm{break}}={39.7}_{-9.1}^{+20.3}\ {M}_{\odot };$ 90% credible bounds on the location of this break are denoted by the gray vertical band in the second panel of Figure 3. For masses above the break, the broken power-law model prefers a significantly steeper power-law slope, from ${\alpha }_{1}={1.58}_{-0.86}^{+0.82}$ before to ${\alpha }_{2}={5.6}_{-2.5}^{+4.1}$ after. Figure 5 shows the joint posterior on ${\alpha }_{1}$ and ${\alpha }_{2}$. We infer that ${\alpha }_{2}\gt {\alpha }_{1}$ at a credibility of $98 \%$. The break aligns with the cutoff ${m}_{\max }$ inferred with GWTC-1 data (Abbott et al. 2019a), and we speculate that the steep drop-off in the merger rate that occurs after m_break may be an imprint of PPSN, which are expected to become important for BH masses around $35\,{M}_{\odot }$ (Heger & Woosley 2002; Heger et al. 2003; Woosley & Heger 2015).

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The power law + peak model (third panel of Figure 3) produces a qualitatively similar fit to the one obtained from the broken power-law model. However, a key feature of the power law + peak model is the Gaussian peak at ${33.1}_{-5.6}^{+4.0}\ {M}_{\odot }$, denoted by the gray vertical band in the third panel of Figure 3. Evidence for a peak can be seen in Figure 6, which shows the posterior for ${\lambda }_{\mathrm{peak}}$, the fraction of systems that belong to the Gaussian component. We see that ${\lambda }_{\mathrm{peak}}=0$ (pure power law) is disfavored. It was envisioned (Talbot & Thrane 2018) that the power-law component of the power law + peak model would terminate in the vicinity of this peak to create a high-mass gap. However, in order to accommodate the most massive binaries in GWTC-2, the power law extends to values of ${m}_{\max }={86}_{-13}^{+12}{M}_{\odot }$.

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While the mass spectrum must extend to these high masses, we find that 99% of primary BH masses lie below ${m}_{99 \% }\sim 60\ {M}_{\odot };$ see Table 3. The astrophysical rate density at ∼$80{M}_{\odot }$ (the primary mass of GW190521) is 2 orders of magnitude lower than the rate density at ∼$40{M}_{\odot }$. However, because of selection effects, the posterior predictive distribution skews to much higher masses, as seen in Figure 4, so that the probability of detecting at least one event with ${m}_{1}\geqslant 80{M}_{\odot }$ after observing 44 BBH events drawn from the power law + peak posterior predictive distribution of Figure 4 is high: $32 \%$.

Table 3. The ${m}_{1 \% }$ and ${m}_{99 \% }$ Credible Intervals (90%) for Various Mass Models and Combinations of Events

Events	Mass Model	${m}_{1 \% }$ (${M}_{\odot }$)	${m}_{99 \% }$ (${M}_{\odot }$)
All confident BBH events, excluding GW190814	Truncated	${6.0}_{-2.1}^{+0.8}$	${65.5}_{-9.6}^{+10.4}$
	Broken power law	${5.6}_{-1.6}^{+1.3}$	${57.8}_{-8.7}^{+12.5}$
	power law + peak	${6.0}_{-1.6}^{+1.0}$	${59.7}_{-12.8}^{+13.9}$
	Multipeak	${6.0}_{-1.6}^{+1.0}$	${66.0}_{-16.4}^{+12.1}$
All confident BBH events, including GW190814	Truncated	${2.4}_{-0.3}^{+0.2}$	${60.3}_{-13.2}^{+10.1}$
	Broken power law	${2.6}_{-0.3}^{+0.5}$	${56.1}_{-9.0}^{+12.5}$
	power law + peak	${2.5}_{-0.3}^{+0.3}$	${57.8}_{-14.5}^{+15.3}$
	Multipeak	${2.6}_{-0.3}^{+0.4}$	${63.8}_{-19.1}^{+11.2}$
All confident BBH events, excluding GW190521 and GW190814	Truncated	${5.8}_{-2.8}^{+0.9}$	${52.3}_{-5.2}^{+8.9}$
	Broken power law	${5.6}_{-1.8}^{+1.3}$	${52.3}_{-5.9}^{+9.5}$
	power law + peak	${5.9}_{-1.5}^{+1.0}$	${52.4}_{-7.0}^{+13.2}$
	Multipeak	${6.2}_{-1.5}^{+0.8}$	${58.9}_{-11.7}^{+11.0}$

Note. These variables are defined such that, among the astrophysical BBH population, 1% of systems have primary masses ${m}_{1}\leqslant {m}_{1 \% }$, while 99% have primary masses ${m}_{1}\leqslant {m}_{99 \% }$. The power law + peak, multipeak, and broken power-law models are preferred over the Truncated model.

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We cannot determine whether the high-mass events of GWTC-2 belong to a distinct subpopulation rather than a high-mass tail of the normal BBH population. An additional subpopulation may be expected if high-mass BHs have a different origin from low-mass ones; for example, if they are the products of hierarchical mergers (Fishbach et al. 2017; Gerosa & Berti 2017; Chatziioannou et al. 2019; Doctor et al. 2020; Kimball et al. 2020a). Using the multipeak model (fourth panel in Figure 3), which allows for a second high-mass Gaussian component at ${m}_{1}\gt 50\ {M}_{\odot }$ in addition to the Gaussian component at ${m}_{1}\lesssim 40\ {M}_{\odot }$, we find that the addition of a second Gaussian peak is not preferred by the data. The multipeak model is mildly disfavored compared to power law + peak, with a ${\mathrm{log}}_{10}{ \mathcal B }=-0.3$ (or Bayes factor ${ \mathcal B }=0.5$) for multipeak relative to power law + peak. Motivated by the hypothesis of hierarchical mergers, we consider a variation of the multipeak model in which the location of the second peak is required to be at twice the value of the first peak, that is, ${\mu }_{m,2}=2{\mu }_{m,1}$ (see Appendix B.4). Studying the $({\mu }_{m,1},{\mu }_{m,2})$ panel of the corner plot in Figure 18, we see that the data mildly prefer the second peak at $\approx 70{M}_{\odot }$, consistent with twice the value of the first peak at $\approx 35{M}_{\odot }$. This "modified multipeak" is mildly preferred over the original version by a Bayes factor of ∼2. A similar conclusion is found using the multispin model; as discussed in Section 5.2, we find hints but no significant evidence for subpopulations with distinct spin distributions. Additional evidence for hierarchical mergers is presented in Kimball et al. (2020b).

Within the framework of the broken power-law, power law + peak, and multipeak models, the most massive event, GW190521, appears to be a normal member of the BBH population in the context of the other GWTC-2 events (see Appendix C.2). The event GW190521 is an outlier if we consider it in the context of GWTC-1 with the Truncated model, but we interpret this as a limitation of the Truncated model (see Figure 2; Abbott et al. 2020f).

The GWTC-1 detections showed that BHs more massive than ∼$45\ {M}_{\odot }$ merge relatively rarely based on simple extrapolations from below $45\ {M}_{\odot }$. With GWTC-2, we are beginning to resolve the shape of the primary BH mass spectrum above $45\ {M}_{\odot }$. The implications are not yet clear, but there are intriguing possibilities. One hypothesis is that the events with ${m}_{1}\gt 45\ {M}_{\odot }$ are simply the high-mass tail of the ordinary BBH population and do not form through a distinct channel. For example, if the lower edge of the PPSN gap can be modeled as a smooth tapering rather than a sharp cutoff, the feature at ${m}_{\mathrm{break}}={39.7}_{-9.1}^{+20.3}$ may represent the onset of pair instability. This explanation may pose challenges to our understanding of stellar evolution, since the pair-instability cutoff of BH masses at ∼$40\ {M}_{\odot }$ is thought to be relatively abrupt (Woosley et al. 2002; Woosley 2017; Farmer et al. 2019), even though its precise location is uncertain (Giacobbo et al. 2017; Spera & Mapelli 2017; Croon et al. 2020; Farmer et al. 2020; Mapelli et al. 2020; Marchant & Moriya 2020; van Son et al. 2020). If the PPSN cutoff is indeed sharp and all observed BBH systems lie below the PPSN gap, the cutoff must occur at relatively high masses; in the Truncated model, ${m}_{\max }={78.5}_{-9.4}^{+14.1}\ {M}_{\odot }$ (or, excluding the most massive event, GW190521, ${m}_{\max }={57.0}_{-6.6}^{+11.9}\ {M}_{\odot }$). This may have significant implications for nuclear (Farmer et al. 2020) and particle (Croon et al. 2020; Ziegler & Freese 2020) physics.

Another hypothesis is that the events with ${m}_{1}\gt 45\ {M}_{\odot }$ constitute a distinct population created, for example, from hierarchical mergers of lower-mass binaries in globular clusters or galactic nuclei (Miller & Hamilton 2002a; Antonini & Rasio 2016; Kimpson et al. 2016; Rodriguez et al. 2018; Yang et al. 2019; Arca Sedda 2020a). Alternatively, the high-mass gap might be populated from low-metallicity stellar mergers in young star clusters, the remnants of which can merge dynamically (Di Carlo et al. 2019, 2020); BH growth through accretion (Roupas & Kazanas 2019; Natarajan 2021; Rice & Zhang 2021; Safarzadeh & Haiman 2020); or Population III stellar remnants with large hydrogen envelopes (Farrell et al. 2021; Liu & Bromm 2020; Tanikawa et al. 2021).

The BBH primary mass distribution exhibits a global maximum between ∼4 and ∼10 M_☉. Figure 7 shows the joint posterior for the ${m}_{\min }$ and ${\delta }_{m}$ parameters inferred using the power law + peak and broken power-law mass models, including only BBH events with ${m}_{2}\gt 3\ {M}_{\odot }$. Recall that, while the Truncated model has a sharp cutoff at ${m}_{\min }$, the remaining models implement a smooth turn-on of width ${\delta }_{m}$ above ${m}_{\min }$, causing the mass spectrum to peak and turn over between ${m}_{\min }$ and ${\delta }_{m}+{m}_{\min }$ (Talbot & Thrane 2018). Using both the broken power-law and power law + peak models, we find that the primary mass spectrum does not decrease monotonically from $3\ {M}_{\odot }$. Rather, it turns over at ${7.8}_{-2.0}^{+1.8}\ {M}_{\odot }$ (power law + peak model) or ${6.02}_{-1.96}^{+0.78}\ {M}_{\odot }$ (broken power-law model). In other words, the mass distribution must turn over at ${m}_{1}\gt 3\ {M}_{\odot }$ with $99.9 \%$ (assuming the power law + peak model) or $98.5 \%$ (broken power-law model) credibility. As seen in Figure 7, if the BH low-mass cutoff is sharp (${\delta }_{m}=0$), then ${m}_{\min }\gtrsim 4\ {M}_{\odot }$. Conversely, if the BH mass spectrum extends below ${m}_{\min }\lesssim 4\ {M}_{\odot }$, an extended turn-on, ${\delta }_{m}\gtrsim 3\ {M}_{\odot }$, is required. These results support the existence of a low-mass gap (or dip) between ∼$2.6\ {M}_{\odot }$ (the secondary mass of GW190814; the most massive component mass observed below $3\ {M}_{\odot }$) and ∼$6\ {M}_{\odot }$, strengthening the results from Fishbach et al. (2020a), although we cannot determine whether the low-mass gap is empty.

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Since our models do not permit additional features beyond a smooth turn-on to a power law at low masses, they struggle to accommodate GW190814 (with secondary mass at ${m}_{2}={2.59}_{0.09}^{+0.08}$). We can see this by comparing the mass distribution inferred from the events with ${m}_{1}\geqslant {m}_{2}\gt 3\,{M}_{\odot }$, discussed above, to the distribution inferred with GW190814. If GW190814 is a BBH system, the minimum BH mass must extend to ${m}_{\min }={2.18}_{-0.16}^{+0.27}\ {M}_{\odot }$ (see the dashed histograms in Figure 8(a)). In Figure 8(b), we show how the inclusion/exclusion of GW190814 affects the shape of the primary mass distribution below ≲$5\ {M}_{\odot }$. We see that the two distributions are inconsistent at the low-mass end, suggesting that there is a feature in the mass distribution between ∼2.6 and ∼$6\,{M}_{\odot }$ that our models cannot capture. This effect can also be seen in the ${m}_{1 \% }$ values inferred with/without GW190814, shown in Table 3. Assuming the Truncated and power law + peak models, the ${m}_{\min }$ posteriors inferred with GW190814 lie at the ${0.25}_{-0.20}^{+0.44}$ and ${0.78}_{-0.69}^{+2.22}$ percentiles, respectively, of the ${m}_{\min }$ posteriors obtained without GW190814. Even using the broken power-law model, which admits greater overlap in the 1D ${m}_{\min }$ posteriors inferred with/without GW190814, the addition of GW190814 significantly shifts the 2D $({\delta }_{m},{m}_{\min })$ posterior and the inferred mass spectrum. Assuming the broken power-law model with GW190814, the mass at which the mass spectrum turns over is shifted down to ${2.59}_{-0.39}^{+0.78}\ {M}_{\odot }$, which is inconsistent with the turnover mass (the low-mass local maximum) inferred without GW190814, ${6.02}_{-1.96}^{+0.78}\ {M}_{\odot }$. This indicates a failure of our models to fit GW190814 together with the BBH systems of GWTC-2. This finding is supported by additional studies described in Appendix C.3. Because GW190814 is a population outlier with respect to the BBH events of GWTC-2 and our choice of models, we exclude it from the analyses here unless otherwise indicated.

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The distribution of mass ratios is broad. The GWTC-1 events are all individually consistent with q = 1. Describing the conditional mass ratio distribution as a power law $p(q| {m}_{1})\propto {q}^{{\beta }_{q}}$, a population analysis of GWTC-1 allowed ${\beta }_{q}=12$ (our maximum prior bound), consistent with a mass ratio distribution sharply peaked at equal-mass pairings (Abbott et al. 2019a; Roulet & Zaldarriaga 2019; Fishbach & Holz 2020b). The first detections of confidently asymmetric systems were seen in GWTC-2: GW190412 (Abbott et al. 2020a) and GW190814 (Abbott et al. 2020b). Excluding GW190814, our reconstruction of the mass ratio distribution is consistent with the results published in Abbott et al. (2020a): ${\beta }_{q}={1.3}_{-1.5}^{+2.4}$ for the power law + peak model and ${\beta }_{q}={1.4}_{-1.5}^{+2.5}$ for the broken power-law model. We rule out distributions that are sharply peaked around q = 1, with ${\beta }_{q}\lt 2.9$ (power law + peak) and ${\beta }_{q}\lt 3.1$ (broken power law) at 90% credibility. However, we also disfavor distributions that prefer unequal-mass pairings, with ${\beta }_{q}\gt 0$ at $92$% (power law + peak) and $94$% (broken power law) credibility. We find that 90% of systems in the underlying population have mass ratios $q\gt {0.26}_{-0.08}^{+0.14}$.

In this subsection, we highlight the results from the Gaussian, default, and multispin models. We fix the redshift distribution to a nonevolving merger rate. The Gaussian and multispin models assume the mass distributions described in Appendices D.2 and D.3, respectively. For the default spin model, we employ the power law + peak mass model, simultaneously fitting the mass and spin distribution as in the previous subsection.

We observe spin-induced general relativistic precession of the orbital plane. As two BHs merge, the morphology of the resulting gravitational waveform depends on their spins. The spin dependence of a gravitational waveform is determined in part by two phenomenological parameters. First, the effective inspiral spin parameter ${\chi }_{\mathrm{eff}}$ quantifies the spin components aligned with the orbital angular momentum (Damour 2001):

$$\begin{eqnarray}&&{\chi }_{\mathrm{eff}}=\displaystyle \frac{{\chi }_{1}\cos {\theta }_{1}+q\,{\chi }_{2}\cos {\theta }_{2}}{1+q}.\end{eqnarray} \tag{ 5 }$$

Here ${\chi }_{1}$ and ${\chi }_{2}$ are the dimensionless component spins, defined by ${\chi }_{i}=| {{cS}}_{i}/({{Gm}}_{i}^{2})|$, where S_i is the spin angular momentum of component i, and ${\theta }_{1}$ and ${\theta }_{2}$ are the misalignment angles between the component spins and the orbital angular momentum. Second, spins with components perpendicular to the orbital angular momentum drive relativistic precession of the orbital plane (Apostolatos et al. 1994). The effect is quantified by the effective precession spin parameter (Schmidt et al. 2012; Hannam et al. 2014; Schmidt et al. 2015):

$$\begin{eqnarray}&&{\chi }_{{\rm{p}}}=\max \left[{\chi }_{1}\sin {\theta }_{1},\left(\displaystyle \frac{4q+3}{4+3q}\right)q\,{\chi }_{2}\sin {\theta }_{2}\right].\end{eqnarray} \tag{ 6 }$$

A nonzero value of ${\chi }_{{\rm{p}}}$ indicates the presence of relativistic spin-induced precession of the orbital plane. Although the component spin tilts ${\theta }_{1}$ and ${\theta }_{2}$ appearing in Equations (5) and (6) generically evolve over the course of a binary inspiral, ${\chi }_{\mathrm{eff}}$ and ${\chi }_{{\rm{p}}}$ are themselves approximately conserved quantities (Kidder 1995; Schmidt et al. 2015).

The first unambiguous measurements of BH spin in gravitational-wave astronomy came from analyses of the BBH event GW151226. This system had ${\chi }_{\mathrm{eff}}\gt 0$ at 99% credibility, with at least one of its components having a spin magnitude $\chi \gt 0.2$ and spin misalignment angles consistent with ${\theta }_{1}={\theta }_{2}=0$ (Abbott et al. 2016b). While analyses of GWTC-1 found no clear evidence for spin in the other events in GWTC-1 (Miller et al. 2020; but also see Zackay et al. 2019 and Huang et al. 2020), GWTC-1 is collectively inconsistent with a population of nonspinning BHs, if one allows for both spinning and nonspinning subpopulations (Kimball et al. 2020a). Moreover, population analyses of GWTC-1 mildly disfavor the scenario in which all spins are perfectly aligned (${\theta }_{1}={\theta }_{2}=0$), although the degree of misalignment is degenerate with the spin magnitude distribution (Farr et al. 2017, 2018; Tiwari et al. 2018; Abbott et al. 2019a; Wysocki et al. 2019b).

In GWTC-2, additional BBH events are observed with confidently positive effective inspiral spin parameters. No individual event is observed with confidently negative ${\chi }_{\mathrm{eff}}$ (Abbott et al. 2020c). Several events, including GW190521 (Abbott et al. 2020e, 2020f) and GW190412 (Abbott et al. 2020a), show moderate evidence for nonzero ${\chi }_{{\rm{p}}}$, but no single event unambiguously exhibits spin-induced precession (Abbott et al. 2020c).

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Using the default model described in Section 3 (see also Appendix D.1), we obtain evidence for nonvanishing spin–orbit misalignment among the population of BBH events in GWTC-2. The default model describes the distribution of spin–orbit misalignments as a mixture of two components: a component with isotropically oriented spins and a preferentially aligned component with $z=\cos \theta$ values centered at z = 0 (perfect alignment) with a Gaussian spread of width ${\sigma }_{t}$. In the left panel of Figure 9, we show the joint posterior on ${\sigma }_{t}$ and the fraction ζ of events in the preferentially aligned subpopulation. Perfect alignment corresponds to $\zeta =1$ and ${\sigma }_{t}=0$. We see that this case is ruled out at >99% credibility. Thus, either a nonzero fraction of BBH events exhibit isotropically oriented spins or BBH spins are preferentially aligned to their orbits but with a nonvanishing spread. Either case constitutes an observation of in-plane spin components among the BBH population. The gray shaded regions in the left panel of Figure 9 show the values of ζ and ${\sigma }_{t}$ that are artificially excluded by the prior and/or finite sampling effects; the true measurement using GWTC-2 lies well away from this artificial exclusion region. ²¹⁹

In Figure 10, discussed further below, we plot the range of component spin magnitude and tilt angle distributions recovered using the default model. Although the data are consistent with tilt angle distributions that favor alignment, distributions that are highly peaked at $\cos {\theta }_{\mathrm{1,2}}=1$ are ruled out.

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A similar conclusion regarding the presence of in-plane spin components may be drawn using the Gaussian spin model, which imposes an entirely different parameterization for the BH spin distribution and makes different assumptions regarding their masses. In particular, when measuring the mean ${\mu }_{p}$ and standard deviation ${\sigma }_{p}$ of the ${\chi }_{{\rm{p}}}$ distribution, the case ${\mu }_{p}={\sigma }_{p}=0$ is ruled out at >99% credibility; fewer than 1% of posterior samples occur at ${\mu }_{p}\leqslant 0.05$ and ${\sigma }_{p}\leqslant 0.05$. Since any nonzero ${\mu }_{p}$ or ${\sigma }_{p}$ implies the existence of spin-induced precession, this result supports the observation of spin misalignment seen in the default model. In the right panel of Figure 9, the blue curve and blue shaded region mark the median and 90% credible bound, respectively, on $p({\chi }_{{\rm{p}}})$ as inferred by the Gaussian model. While the blue shaded region in this figure suggests a ${\chi }_{{\rm{p}}}$ distribution that peaks at ∼0.2, there are in fact two morphologies preferred by the data according to the Gaussian model: the recovered ${\chi }_{{\rm{p}}}$ distribution is either broad or narrowly peaked at ${\chi }_{{\rm{p}}}\approx 0.2$. This is illustrated by the inset, in which we plot an ensemble of distributions corresponding to individual draws from the $({\mu }_{p},{\sigma }_{p})$ posterior; the dashed black curves highlight traces representative of the two permitted morphologies.

For comparison, the orange curve in the right panel of Figure 9 shows the ${\chi }_{{\rm{p}}}$ distribution implied by the default results discussed above. There are several potentially meaningful differences between the results from the Gaussian and default models. In particular, the default model predicts ${\chi }_{{\rm{p}}}$ distributions that are generally broader and peaked at lower values. This is due to additional physical constraints imposed by the default spin model; component spins are presumed to preferentially cluster about $\theta =0$, an assumption that preferentially favors smaller ${\chi }_{{\rm{p}}}$ values. Nevertheless, the two models agree well within the statistical uncertainties, ²²⁰ indicating that the identification of spin-induced precession is robust to the systematic modeling choices and prior uncertainties.

As mentioned above, GW190521 and GW190412 individually show mild evidence of precession, with the ${\chi }_{{\rm{p}}}$ posteriors shifted away from their respective priors (Abbott et al. 2020a, 2020e, 2020f). To verify that our population-level conclusions are not driven primarily by these two events, we have repeated the Gaussian analysis excluding GW190521 and GW190412. Our results again exclude ${\mu }_{p}={\sigma }_{p}=0$ at a similar level of confidence (>99% credibility). This implies that the signature of precession observed here is due to the combined influence of many systems with only weakly measured ${\chi }_{{\rm{p}}}$, consistent with expectations from simulation studies (Fairhurst et al. 2020; Wysocki et al. 2019b).