<table><caption>**Table 11**
Summary of MULTISPIN Parameters</caption><thead><tr><th>Parameter</th><th>Description</th><th>Prior</th></tr></thead><tbody><tr><td>R<sub>pl</sub></td><td>Local merger rate for the low-mass power-law subpopulation</td><td>U(0, 5000)</td></tr><tr><td>R<sub>g</sub></td><td>Local merger rate for the high-mass Gaussian subpopulation</td><td>U(0, 5000)</td></tr><tr><td>α<sub>m</sub></td><td>Power-law slope of the primary mass distribution for the low-mass subpopulation</td><td>U(-4, 12)</td></tr><tr><td>β<sub>q</sub></td><td>Power-law slope of the mass ratio distribution for the low-mass subpopulation</td><td>U(-4, 10)</td></tr><tr><td>m<sub>min</sub></td><td>Minimum mass of the primary mass distribution for the low-mass subpopulation</td><td>U(2, 10)</td></tr><tr><td>m<sub>max</sub></td><td>Maximum mass of the primary mass distribution for the low-mass subpopulation</td><td>U(30, 100)</td></tr><tr><td>μ<sub>m1</sub></td><td>Centroid of the primary mass distribution for the high-mass subpopulation</td><td>U(20, 50)</td></tr><tr><td>σ<sub>m1</sub></td><td>Width of the primary mass distribution for the high-mass subpopulation</td><td>U(0.4, 10)</td></tr><tr><td>μ<sub>m2</sub></td><td>Centroid of the secondary mass distribution for the high-mass subpopulation</td><td>U(20, 50)</td></tr><tr><td>σ<sub>m2</sub></td><td>Width of the secondary mass distribution for the high-mass subpopulation</td><td>U(0.4, 10)</td></tr><tr><td>Meanχ<sub>1,pl</sub></td><td>Mean of the beta distribution of primary spin magnitudes for the low-mass subpopulation</td><td>U(0, 1)</td></tr><tr><td>Varχ<sub>1,pl</sub></td><td>Variance of the beta distribution of primary spin magnitudes for the low-mass subpopulation</td><td>U(0, 0.25)</td></tr><tr><td>σ<sub>1,pl</sub></td><td>Width of the TRUNCATED Gaussian distribution of the cosine of the primary spin-tilt angle for the low-mass subpopulation</td><td>U(0, 4)</td></tr><tr><td>Meanχ<sub>2,pl</sub></td><td>Mean of the beta distribution of secondary spin magnitudes for the low-mass subpopulation</td><td>U(0, 1)</td></tr><tr><td>Varχ<sub>2,pl</sub></td><td>Variance of the beta distribution of secondary spin magnitudes for the low-mass subpopulation</td><td>U(0, 0.25)</td></tr><tr><td>σ<sub>2,pl</sub></td><td>Width of the TRUNCATED Gaussian distribution of cos(secondary spin-tilt angle) for the low-mass subpopulation</td><td>U(0, 4)</td></tr><tr><td>Meanχ<sub>1,g</sub></td><td>Mean of the beta distribution of primary spin magnitudes for the high-mass subpopulation</td><td>U(0, 1)</td></tr><tr><td>Varχ<sub>1,g</sub></td><td>Variance of the beta distribution of primary spin magnitudes for the high-mass subpopulation</td><td>U(0, 0.25)</td></tr><tr><td>σ<sub>1,g</sub></td><td>Width of the TRUNCATED Gaussian distribution of cos(secondary spin-tilt angle) for the high-mass subpopulation</td><td>U(0, 4)</td></tr><tr><td>Meanχ<sub>2,g</sub></td><td>Mean of the beta distribution of secondary spin magnitudes for the high-mass subpopulation</td><td>U(0, 1)</td></tr><tr><td>Varχ<sub>2,g</sub></td><td>Variance of the beta distribution of secondary spin magnitudes for the high-mass subpopulation</td><td>U(0, 0.25)</td></tr><tr><td>σ<sub>2,g</sub></td><td>Width of the TRUNCATED Gaussian distribution of cos(secondary spin-tilt angle) for the high-mass subpopulation</td><td>U(0, 4)</td></tr></tbody></table>

### D.3. MULTISPIN Model

This model is an extension of the TRUNCATED mass model with an additional Gaussian component. It is similar to the POWER LAW + PEAK model, but there are several differences. First, the high-mass subpopulation in the MULTISPIN model is described by a Gaussian in both $m_1$ and $m_2$ (up to the $m_1 \ge m_2$ truncation), while POWER LAW + PEAK only models $m_1$ as a Gaussian and assumes that all BBH systems are described by a power-law distribution in mass ratio $q$. Most importantly, as its name suggests, the MULTISPIN model allows each subpopulation to have its own independent spin distribution, each of which follows the DEFAULT model, with $\zeta = 1$. This allows us to probe whether the spin distribution varies with mass. The parameters for MULTISPIN are summarized in Table 11.

## Appendix E
Redshift Evolution Models

The POWER-LAW redshift evolution model parameterizes the merger rate density as

$$ \mathcal{R}(z) = \mathcal{R}_0(1 + z)^{\kappa}, \qquad (E1) $$

where $\mathcal{R}_0$ denotes the merger rate density at $z = 0$. This implies that the redshift distribution is

$$ \frac{dN}{dz} = C \frac{dV_c}{dz} (1 + z)^{\kappa - 1}, \qquad (E2) $$

where $dV_c/dz$ is the differential comoving volume, and $C$ is related to $\mathcal{R}_0$ by

$$ \mathcal{R}_0 = C \frac{dV_c}{dz} \left[ \int_0^{z_{\text{max}}} \frac{dV_c}{dz} (1+z)^{\kappa-1} \right]^{-1}. \qquad (E3) $$

<figure><img src="image_33.png" /><figcaption>**Figure 30.** Posterior predictive check: the cumulative density function for the POWER-LAW EVOLUTION model. The model is a good fit to the data. This analysis excludes GW190814.</figcaption></figure>

We take $z_{\text{max}} = 2.3$ in the analysis, as this is a conservative upper bound on the redshift at which we could detect BBH systems during O3a within the mass range considered here. When fitting this model, we employ a uniform prior on $\kappa$ centered at $\kappa = 0$. The value $\kappa = 0$ corresponds to no evolution, i.e., a merger rate that is uniform in comoving volume and source-frame time. We take a sufficiently wide prior so that the likelihood is entirely within the prior range, $\kappa \in (-6, 6)$. We show a posterior predictive check for this redshift evolution model in Figure 30.
## Appendix F

### Gravitational-wave Lensing

It has been suggested that gravitational-wave lensing could bias the estimate of binary masses (Dai et al. 2017; Li et al. 2018; Ng et al. 2018b; Oguri 2018; Hannuksela et al. 2019), which could lead to a biased population inference. However, based on the predictions on the number of expected gravitational-wave sources and the distribution of galaxy lenses in the universe, Li et al. (2018) and Oguri (2018) predicted that only around one in a thousand observed events are lensed, although this estimate can vary depending on the redshift evolution of the merger rate density. The lensing rate is expected to be similarly rare for galaxy clusters (Smith et al. 2018). As the expected lensing rate is low compared to the number of events in GWTC-2, we assume that all events in our sample are unlensed.

## ORCID iDs

K. Ackley https://orcid.org/0000-0002-8648-0767
K. Agatsuma https://orcid.org/0000-0002-3952-5985
S. Anand https://orcid.org/0000-0003-3768-7515
G. Ashton https://orcid.org/0000-0001-7288-2231
F. Aubin https://orcid.org/0000-0002-8241-4156
F. Badaracco https://orcid.org/0000-0001-8553-7904
S. Banagiri https://orcid.org/0000-0001-7852-7484
I. Bartos https://orcid.org/0000-0001-5607-3637
B. Bécsy https://orcid.org/0000-0003-0909-5563
M. Bejger https://orcid.org/0000-0002-4991-8213
S. Bernuzzi https://orcid.org/0000-0002-2334-0935
D. Bhattacharjee https://orcid.org/0000-0003-0466-750X
I. A. Bilenko https://orcid.org/0000-0002-9543-0542
M. Boer https://orcid.org/0000-0001-9157-4349
Y. Bouffanais https://orcid.org/0000-0003-3462-0366
F. Brighenti https://orcid.org/0000-0001-9807-8479
R. Buscicchio https://orcid.org/0000-0002-7387-6754
M. Cabero https://orcid.org/0000-0003-4059-4512
M. W. Coughlin https://orcid.org/0000-0002-8262-2924
M. Fishbach https://orcid.org/0000-0002-1980-5293
D. E. Holz https://orcid.org/0000-0002-0175-5064
V. Kalogera https://orcid.org/0000-0001-9236-5469
S. Nissanke https://orcid.org/0000-0001-6573-7773
G. Raaijmakers https://orcid.org/0000-0002-9397-786X

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