Issue 
A&A
Volume 668, December 2022



Article Number  L2  
Number of page(s)  16  
Section  Letters to the Editor  
DOI  https://doi.org/10.1051/00046361/202245084  
Published online  09 December 2022 
Letter to the Editor
Spin it as you like: The (lack of a) measurement of the spin tilt distribution with LIGOVirgoKAGRA binary black holes
^{1}
LIGO Laboratory, Massachusetts Institute of Technology, 185 Albany St, Cambridge, MA 02139, USA
^{2}
Department of Physics and Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge, MA 02139, USA
email: salvo@mit.edu
Received:
29
September
2022
Accepted:
17
November
2022
Context. The growing set of gravitationalwave sources is being used to measure the properties of the underlying astrophysical populations of compact objects, black holes, and neutron stars. Most of the detected systems are black hole binaries. While much has been learned about black holes by analyzing the latest LIGOVirgoKAGRA (LVK) catalog, GWTC3, a measurement of the astrophysical distribution of the black hole spin orientations remains elusive. This is usually probed by measuring the cosine of the tilt angle (cosτ) between each black hole spin and the orbital angular momentum, with cosτ = +1 being perfect alignment.
Aims. The LVK Collaboration has modeled the cosτ distribution as a mixture of an isotropic component and a Gaussian component with mean fixed at +1 and width measured from the data. We want to verify if the data require the existence of such a peak at cosτ = +1.
Methods. We used various alternative models for the astrophysical tilt distribution and measured their parameters using the LVK GWTC3 catalog.
Results. We find that (a) augmenting the LVK model, such that the mean μ of the Gaussian is not fixed at +1, returns results that strongly depend on priors. If we allow μ > +1, then the resulting astrophysical cosτ distribution peaks at +1 and looks linear, rather than Gaussian. If we constrain −1 ≤ μ ≤ +1, the Gaussian component peaks at μ = 0.48_{−0.99}^{+0.46} (median and 90% symmetric credible interval). Two other twocomponent mixture models yield cosτ distributions that either have a broad peak centered at 0.19_{−0.18}^{+0.22} or a plateau that spans the range [ − 0.5, +1], without a clear peak at +1. (b) All of the models we considered agree as to there being no excess of black hole tilts at around −1. (c) While yielding quite different posteriors, the models considered in this work have Bayesian evidences that are the same within error bars.
Conclusions. We conclude that the current dataset is not sufficiently informative to draw any modelindependent conclusions on the astrophysical distribution of spin tilts, except that there is no excess of spins with negatively aligned tilts.
Key words: gravitational waves / black hole physics / methods: data analysis
© S. Vitale et al. 2022
Open Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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1. Introduction
More than 90 binary black holes (BBHs) have been detected in the data of the groundbased gravitationalwave detectors LIGO (Aasi et al. 2015) and Virgo (Acernese et al. 2015) by the LIGOVirgoKagra (LVK) collaboration and other groups (Abbott et al. 2021b; Nitz et al. 2021; Olsen et al. 2022). This dataset has been used to infer the properties of the underlying population – or populations – of BBHs. Among the parameters of interest, the masses and spins of the black holes play a prominent role since they can shed light on the binary formation channels^{1} (e.g., Vitale et al. 2017; Farr et al. 2017; Zevin et al. 2017, 2021; Farr et al. 2018; Wong et al. 2021; Bouffanais et al. 2021).
There currently exist a few approaches toward measuring the population properties. The first is to use a functional form for a reasonable population distribution, parameterized by some phenomenological parameters. For example, the LVK parameterized the primary mass distribution of the black holes as a mixture of a power law distribution and a Gaussian component (Abbott et al. 2021e; Talbot & Thrane 2018; Fishbach & Holz 2017). This model includes several hyperparameters (since they pertain to the population as a whole, not to the individual events), which are measured from the data: the slope of the power law, the minimum and maximum black hole mass (including a smoothing parameter), the mean and standard deviation of the Gaussian component, and the branching ratio between the power law and Gaussian components. It is worth noting that not all parametric models make equally strong assumptions: an example of a more flexible parametrization is the beta distribution that the LVK has used to describe the population distribution of black hole spin magnitudes (Wysocki et al. 2019). Those more elastic models might be more appropriate if one does not have strong observational or theoretical expectations about what the astrophysical distribution of a parameter should look like, or simply if they want to be more conservative. Just as Bayesian priors can significantly affect the posterior when the likelihood does not have a strong peak when analyzing individual compact binary coalescences, a hypermodel that is too strong could leave imprints on the inferred hyperparameters.
A second approach is to use nonparametric models, based on Gaussian processes, for example (Tiwari 2021; Edelman et al. 2022; Rinaldi & Del Pozzo 2021; Mandel et al. 2017; Vitale et al. 2019). Those usually have many more free parameters, which allows them to fit features in the data that parameterized models might miss. However, their larger number of parameters implies they might need more sources to reach a level of precision comparable to parametric models. Ideally, when strong parametric models are used, one would like to check that the results are not impacted by the model itself, and instead reveal features that are genuinely present in the data. A possible approach is to run multiple models (parametric and nonparametric) and verify that they agree to within statistical uncertainties.
Finally, recent work has focused on using the predictions of numerical simulations as a model. This typically involves applying machine learning (Wong et al. 2021) or density estimation techniques (Zevin et al. 2017; Bouffanais et al. 2021) to the binary parameter distributions’ output by rapid population synthesis codes or Nbody simulations. While this approach is more astrophysically motivated, the population synthesis simulations have their own uncertainties and assumptions and often include so many free parameters, such that a complete exploration of the model space is not possible with current computational techniques (e.g., Broekgaarden et al. 2021). The impact of modeling on astrophysical inference of gravitationalwave sources has already become apparent in recent months. Several groups have investigated whether there is evidence for a fraction of black hole spin magnitude to be vanishingly small, finding results that depend on the model to a large extent (Callister et al. 2022; Galaudage et al. 2021; Tong et al. 2022; Roulet et al. 2021; Mould et al. 2022). We note that Callister et al. (2022) also provide a comprehensive summary of the status of that measurement.
In this Letter we focus on the inference of the population distribution of tilt angles for the black hole binaries in the latest LVK catalog. This is the angle that each of the black hole spin vectors forms with the orbital angular momentum at some reference frequency (following the LVK data release, we use 20 Hz for the reference frequency^{2}). In their latest catalog of BBHs, the LVK collaboration characterized this distribution by using a mixture model composed of an isotropic distribution plus a Gaussian distribution that peaks at cosτ = 1 (i.e., when the spin vector and the angular momentum are aligned) with an unknown width (Talbot & Thrane 2017; Abbott et al. 2021e). This model reflects expectations from astrophysical binary modeling. Indeed, numerical simulations suggest that binaries that formed in galactic fields should have spins preferentially aligned with the angular momentum (e.g., Tutukov & Yungelson 1993; Kalogera 2000; Belczynski et al. 2020; Zaldarriaga et al. 2018; Stevenson et al. 2017; Gerosa et al. 2018), whereas binaries that formed dynamically (i.e., in globular or star clusters) should have randomly oriented tilts (e.g., Portegies Zwart & McMillan 2002; Rodriguez et al. 2015; Antonini & Rasio 2016; Rodriguez et al. 2019; Gerosa & Fishbach 2021).
While this is a reasonable model, we are interested in verifying whether it is actually supported by the available data, or whether we are instead getting posteriors that are driven by that model. Callister et al. (2022) and Tong et al. (2022) recently considered alternative models for the tilt angles, but they focused on whether there is a cutoff at negative cosine tilts (i.e., for antialigned spins), and if that answer depends on the model for the spin magnitude. On the other hand, we do not limit our investigation to the existence of negative tilts, but instead are interested in what–if anything–can be said about the tilt inference that is not strongly dependent on the model being used. We have considered different alternative models^{3} and verified that all of them yield Bayesian evidences (and maximum log likelihood values), which are comparable with the default model used by the LVK. We have also included models that allow for a correlation between cosτ and other parameters (binary mass, mass ratio, or spin magnitude, in turn) and find that those too yield similar evidences. Critically, all of these alternative – and equally supported by the data – models yield noticeably different posterior distributions for the tilt population relative to the default LVK model. In particular, different models give different support to the existence and position of a feature at positive cosτ. On the other hand, they all agree as to there being no excess of systems with cosτ ≃ −1. We conclude that the current constraints on the distribution of tilt angles are significantly affected by the model used, and that more sources (or weaker models) are needed before any conclusions can be drawn about the astrophysical distribution of BBH tilts.
2. Methods
We used hierarchical Bayesian analysis – extensively described in Appendix A – to infer the astrophysical properties of the BBHs reported in GWTC3. To represent the astrophysical distribution of primary mass, mass ratio, redshift, and spin magnitudes, we used the flagship models used by the LVK in Abbott et al. (2021e). Namely, the primary mass distribution is their “Power law + Peak” (Talbot & Thrane 2018); the mass ratio is a power law (Fishbach & Holz 2020); the two spin magnitudes are independently and identically distributed according to a beta distribution (Wysocki et al. 2019); and the redshift is evolving with a power law (Fishbach et al. 2018). For the (cosine^{4}) tilt distributions, we considered several models of increasing complexity.
3. Results
In Appendix B we report our results when we used, for the tilt distribution, the “DEFAULT” spin model of Appendix B2a of Abbott et al. (2021e) (referred to as LVK default in the rest of the paper): this serves as a useful comparison for the more complex models described in the reminder of this work. In this section we reanalyze the GWTC3 BBH with different parameterized twocomponent mixture models for cosτ. We report results for threecomponent models in Appendix C. It is assumed that the cosτ distribution is not correlated to any other of the astrophysical parameters. This assumption is revisited in Appendix D.
3.1. Isotropic + Gaussian model
To check if the data require that the Gaussian component of the LVK default mixture model peaks at cosτ = 1 (Appendix B), we relaxed the assumption that the normal distribution must be centered at +1, that is to say we treated the mean of the Gaussian component μ as another model parameter:
The prior for μ is uniform in the range [ − 1, 1] (Table G.2 reports the priors for the hyperparameters of all models used in the paper).
The top panel of Fig. 1 shows the resulting posterior for the tilt angle. For the mean of the Gaussian component, we measured (unless otherwise stated, we quote the median and 90% symmetric credible interval). Some of the uncertainty in this measurement is due to our choice to allow for large σ, since Gaussians with large σ are rather flat and can be centered anywhere without significantly affecting the likelihood. If we restrict the prior space to only allow for narrower Gaussians, then μ is much more constrained. For example, if we only keep samples with σ < 0.5 (σ < 1), then (). This can also be seen in a corner plot of the mean and standard deviation of the Gaussian component (Fig. 2, dark green). The data prefer positive means with standard deviations in the approximate range σ ∈ [0.25, 1.5]. Smaller values of σ are excluded, as are Gaussians centered at negative values of μ, in other words such that χ_{eff} – the massweighted projection of the total spin along the angular momentum (Damour 2001) – would be negative. The bottom panel of Fig. 1 shows the posterior on the differential merger rate per unit cosτ for the same model.
Fig. 1. Posterior for cosτ (top panel) and differential merger rate per unit cosτ (bottom panel) obtained using the Isotropic + Gaussian model when the mean of the Gaussian component is allowed to vary in the range μ ∈ [ − 1, 1]. The two thin black dotted lines in the top panel show the 90% CI obtained by drawing the model’s hyperparameters from their priors. In both panels, the thin black lines represent individual posterior draws, whereas the colored band shows the 90% CI. The thick dashed line within the band is the median. 
It is worth noting that the astrophysical cosτ posterior changes entirely if the prior for μ is extended to allow for values outside of the range [ − 1, 1]; whereas, the resulting population model is, of course, still truncated and properly normalized in that domain. For example, Fig. 3 shows the posterior for the tilt distribution obtained with a wider μ prior, uniform in the range [ − 5, 5]. Here again it is the case that the distribution is consistent with having a peak for aligned spins. A look at the joint distribution of μ and σ, Fig. 2, reveals that the peak at μ = 1 is not obtained because the Gaussian component peaks there, but rather by truncating Gaussians that peak at μ > 1 and have large standard deviations. This results in a much steeper shape for the posterior near the cosτ = +1 edge than what could possibly be obtained by forcing −1 ≤ μ ≤ 1. The only feature that seems solid against model variations is the fact that there is no excess of systems at negative cosτ.
Fig. 2. Joint and marginal posteriors for the mean and standard deviation of the Gaussian component for the Isotropic + Gaussian model, as well as for the branching ratio 𝔤, when the mean of the Gaussian component is allowed to vary in the range μ ∈ [ − 1, 1] (dark green) or μ ∈ [ − 5, 5] (light green). 
We can better visualize what happens at the edges of the cosτ domain – that is for values of tilts close to aligned (cosτ ≃ 1) or antialigned (cosτ ≃ −1) – by plotting the ratio of the posterior support for aligned spin versus antialigned spins. This is equivalent to making a histogram of the ratio of the value that the thin black curves in Fig. 3 take on the far right and far left side. Specifically, for each of the posterior draws, we calculated an asymmetry coefficient Y defined as
Fig. 3. Same as Fig. 1, but for the Isotropic + Gaussian model, when the mean of the Gaussian component is allowed to vary in the range μ ∈ [ − 5, 5]. 
and plotted the resulting histogram. When Y = 1, the cosτ distribution takes the same value at both of the edges; when Y > 1 (Y < 1), the cosτ distribution has more support for aligned (antialigned) systems than for antialigned (aligned) ones. We notice that the LVK default model excludes an a priori excess of antialigned tilts, that is Y < 1. This is instead not true for the other models we consider in this section. This is shown in the top panel of Fig. 4, for δ = 0.01 (i.e., considering ∼8 degree widths around ±L, where L is the angular momentum vector). We see that the prior (dashed green curve) of Y for the Isotropic + Gaussian model can extend to values smaller than 1, that is it can produce more antialigned spins than aligned ones (readers can compare this with the blue dashed curve and Appendix B). In fact, we see that the prior for this asymmetry probe is much less strong than in the default LVK default model as it does not exclude Y < 1. As for the LVK default, the posterior of Y is not inconsistent with 1. Values of Y smaller than 1 – that is an excess of antialigned spins – are severely suppressed relative to the prior, and so is a large excess of positive tilts. The posterior for Y has a rather broad peak in the range ∼[1, 2], corresponding to distributions that are consistent with being either isotropic or having a mild excess of positive alignment. In Appendix F we show similar plots for different values of δ.
Fig. 4. Unnormalized histogram of Y(δ), the ratio between the probability of cosτ for cosτ ∈ [1 − δ, 1] and cosτ ∈ [ − 1, −1 + δ] for the models of Sects. 3.1–3.3 for δ = 0.01. We also show the result for the LVK default model (Appendix B) for comparison. The solid curves were obtained by sampling the hyperposteriors. The dashed line reports the same quantity, but drawing the model’s hyperparameters from their priors. Values of Y > 1 imply more support for aligned than antialigned spins. 
In the top panel of Fig. 5, we show the marginalized posteriors of the branching ratio for the isotropic component, 𝔦, of the models described in this section, together with the reference LVK default model. While small variations exist, they all have support across the whole prior range, with a preference for small values of 𝔦. By comparing Figs. 1, 3 and 5, one may be surprised that the branching ratio posteriors for the two Isotropic + Gaussian runs are basically the same, and yet Fig. 1 seems to have a higher density of horizontal (i.e., isotropic) curves. This happens because the Gaussian component becomes isotropic for small means μ and large standard deviations σ. Comparing the two distributions for μ in the topleft panel of Fig. 2, we see that indeed the run where μ is constrained to [ − 1, 1] has more support at small means μ, which, together with large standard deviations σ, yield nearly horizontal posteriors in Fig. 1. To a different extent, the same is true for the models described below: there exist corners of the parameter space where the nonisotropic component can in fact generate a flat distribution, and unless otherwise said we do not a priori exclude that possibility.
Fig. 5. Marginalized posterior for the branching ratio of the isotropic component – 𝔦 – for all of the uncorrelated twocomponent models (Sects. 3.1–3.3). The figure is split into two panels to enhance clarity. In both panels, we also report the posterior obtained with the LVK default model (yellow dashed line) for comparison. 
3.2. Isotropic + Beta model
To allow for a more elastic model for the nonisotropic component, we now replace the Gaussian component of the previous section with a beta distribution. We have:
We offset the input of the beta distribution and scaled its maximum value such that it spans the domain [ − 1, 1]. We stress that we did not limit the range of α and β to nonsingular values, that is we did allow them to be smaller than 1 (Table G.2). In turn, this implies that we can get posteriors for cosτ that peak at the edges of the range. The resulting posterior for cosτ is shown in Fig. 6, which also shows, for comparison, the 90% credible interval (CI) when drawing hyperparameters from their priors (thin dashed lines). With this model, we recovered a broad peak at small positive values of cosτ. One can convert the α and β parameters of our rescaled beta distribution to the corresponding mean as
Fig. 6. Same as Fig. 1, but for the Isotropic + Beta model. It is important to note the different scale for the y axis of the bottom panel compared with similar figures for other models. 
We find . While some of the posterior draws do peak at +1, overall the upper edge of the 90% CI band does not show a peak in that region. Compared to what has been seen in the previous section, this model finds more support for for antialigned tilts, with a median value that is at the lower edge of the 90% CI for the Isotropic + Gaussian models. In the bottom panel of Fig. 4, we show in red the posterior (solid line) and prior (dashed line) of the asymmetry coefficient Y defined in Eq. (2). Also for this model, we observed that the posterior disfavors configurations with an excess of antialigned black hole tilts, relative to the prior. We noticed that for the Isotropic + Beta model, values of Y > 1 do not necessarily represent a peak at positive tilts (see Fig. 6), but only that negative tilts are even more suppressed.
The marginalized posterior for the branching ratio of the isotropic component is shown in the bottom panel of Fig. 5 (histogram with dotted hatches). We find that, unlike the Gaussianbased models of Sect. 3.1 or the LVK default model, it does feature a very broad peak in the middle of the range.
3.3. Isotropic + Tukey model
We concluded our exploration of twocomponent models with a mixture of an isotropic distribution and a distribution based on the Tukey window function. Mathematically,
where the exact expression for the functional form of the window function and a few examples are given in Appendix E. Figure 7 reports the resulting posterior distribution for cosτ, together with the prior (dotted black lines). The 90% CI band features a plateau that extends from cosτ ≃ −0.5 to +1.
Fig. 7. Same as Fig. 1, but for the Isotropic + Tukey model. Individual posterior draws are colored according to the corresponding value of the branching ratio for the Tukey component, 𝔱. 
In Fig. 8, we show the posterior distribution for the parameters controlling the Tukey channel, together with the corresponding branching ratio. The Tukey component is centered at T_{x0}, whose fifth and 95th percentile are −0.37 and 0.94, respectively. The marginal posterior for T_{k} prefers values close to 1.8, implying wider Tukey distributions. Smaller values of T_{k} are possible only for small 𝔱, as is expected given that for small 𝔱the data cannot constrain the Tukey component, and the posterior must then resemble the uniform prior, which includes small T_{k}. The posteriors of these two parameters are correlated such that when 𝔱 is large, T_{k} is also large, meaning the resulting cosτ distribution more closely resembles an isotropic distribution. However, when T_{k} gets larger than ∼2, then 𝔱is not constrained at all. This happens because when T_{k} is that large, the Tukey component is extremely close to an isotropic distribution, at which point the whole model is isotropic, and the branching ratio stops being a meaningful parameter. Finally, the posterior for T_{r} is wide, with a preference for larger values, implying a Tukey distribution that ramps up and down smoothly rather than producing sharp features.
Fig. 8. Joint and marginal posteriors for the hyperparameters and branching ratio of the Tukey component in the Isotropic + Tukey model. 
Figure 8 also reveals that large values of T_{k} are responsible for the near entirety of the support at T_{x0} < 0, since a Tukey that is basically a uniform distribution can be centered anywhere without affecting the likelihood. If we restrict the analysis to T_{k} ≤ 2 (T_{k} ≤ 1), we get can see that T_{x0} is much better constrained to (), which excludes negative values at ∼90% credibility. The fact that our generous hyperparameter priors allow for Tukey distributions that resemble isotropic ones also explain the peak at Y = 1 in the bottom panel of Fig. 4 (purple lines, solid for the posterior and dashed for the prior). We find, once again, that the data do not exclude that the cosτ distribution is in fact isotropic, and the only solid conclusion one can make seems to be that there is no excess of systems at cosτ ≃ −1, since p(Y) is heavily suppressed relative to its prior for Y ≲ 1.
4. Conclusions
In this paper we have reanalyzed the LVK’s 69 BBHs of GWTC3, using different models for the astrophysical distribution of the black hole spin tilt angle, in other words the angle the spin vector forms with the orbital angular momentum at a reference frequency (20 Hz). Black hole spin tilts can yield precious information about their astrophysical formation channels. It is usually expected that dynamical formation of binaries results in an isotropic distribution of the spin vectors (e.g., Portegies Zwart & McMillan 2002; Rodriguez et al. 2015; Antonini & Rasio 2016; Rodriguez et al. 2019; Gerosa & Fishbach 2021). On the other hand, for black hole binaries that formed in the field via isolated binary evolution, it is expected that the spins are nearly aligned with the angular momentum (e.g., Tutukov & Yungelson 1993; Kalogera 2000; Belczynski et al. 2020; Zaldarriaga et al. 2018; Stevenson et al. 2017; Gerosa et al. 2018), that is to say that tilts are small, as the angular momenta of the progenitor stars are aligned by starstar and stardisk interactions (Hut 1981; Packet 1981). Indeed, if the binary forms in the field, the only mechanism that could yield significant black hole tilts are asymmetries in the supernovae explosions that create the black holes. These asymmetries can impart a natal kick large enough to tilt the orbital plane (Katz 1975; Kalogera 2000; Hurley et al. 2002). However, the black hole natal kick distribution is poorly understood both theoretically (e.g., Dominik et al. 2012; Zevin et al. 2017; Mapelli & Giacobbo 2018; Repetto et al. 2012; Giacobbo & Mapelli 2020; Fragos et al. 2010) and observationally (Brandt et al. 1995; Nelemans et al. 1999; Mirabel et al. 2001, 2002; Wong et al. 2014).
These expectations explain why in their most recent catalog, the LVK has modeled the astrophysical tilt distribution as a mixture of two components: an isotropic part and a Gaussian distribution centered at cosτ = 1 and with a width that is measured from the data. This is a rather strong model, as it forces a Gaussian onto the data that must be centered at +1. This might not be advisable because individual tilt measurements are usually broad, which implies that the functional form of the assumed astrophysical model can leave a discernible imprint on the posterior. Given the rather large uncertainties as to how much spin misalignment can be produced in each channel, and the fact that the BBH population currently available might contain back holes that formed in different channels (Zevin et al. 2021; Wong et al. 2021; Bouffanais et al. 2021; Franciolini & Pani 2022), it is legitimate to question whether the data require that the cosτ distribution peaks at +1, as opposed to just being consistent with it. We find that it does not.
We consider three twocomponent mixture models: Isotropic + Gaussian, made of an isotropic component and a Gaussian component whose mean is not fixed at +1, but rather measured from the data; Isotropic + Beta, made of an isotropic component and a (singular) beta component; and Isotropic + Tukey, made of an isotropic component and a distribution based on the Tukey window. We find that the only model with a posterior that peaks at cosτ = 1 is the Isotropic + Gaussian, and only if we allow the mean of the Gaussian to take values outside of the cosτ domain [ − 1, 1]: Fig. 2 shows that in this case the Gaussian component peaks at μ > 1 and is very broad, yielding a peak at +1 (Fig. 3). The other two models yield either a posterior that peaks at cosτ ≃ 0.2 and no peak at +1, or a plateau from cosτ ≃ −0.5 to +1. For all of the above models, the data do not decisively rule out a fully isotropic tilt distribution, but they are inconsistent with an excess of systems with large and negative tilts. The models we considered found features–in addition to an isotropic distribution–whose exact shape depends on the model’s flexibility to a large extent. Therefore, the fact that the LVK default model finds support for a peak at cosτ = 1 can be explained because it can only add support at cosτ = 1 when trying to match any population features in addition to the isotropic distribution. Our results agree with previous results for population models fitting the distribution of χ_{eff}, which find only a small fraction of sources with negative χ_{eff}, implying negative tilts (Roulet & Zaldarriaga 2019; Miller et al. 2020; Roulet et al. 2021; Callister et al. 2022). Indeed, if we recast our inferred distributions for cosτ and spin magnitude to the resulting χ_{eff} distribution, we would obtain results consistent with Abbott et al. (2021e). In the appendices below, we report on other models. In Appendix C we consider three threecomponent mixture models: Isotropic + 2 Gaussians, made of an isotropic component and two Gaussian components; Isotropic + Gaussian + Beta, made of an isotropic component a (nonsingular) beta component and a Gaussian component; and Isotropic + Gaussian + Tukey, made of an isotropic component, a distribution based on the Tukey window and a Gaussian component. These more elastic models are consistent with a broad plateau in the cosτ posterior that extends from ∼ − 0.5 to +1. Whether there are also peaks or features at small positive cosτ and/or at +1 depends on the exact model.
In Appendix D we augment some of the twocomponent mixture models to allow for correlations between the tilt angles and another of the binary parameters: component masses, component spins, mass ratio, and total mass in turn. One might expect some correlation as the mechanisms that can misalign the binary orbital plane or the black hole spins are affected by the binary parameters (e.g., Janka & Mueller 1994; Burrows & Hayes 1996; Fryer & Kusenko 2006; Gerosa & Fishbach 2021). To keep the number of model parameters limited – consistent with the relatively limited number of sources – we only consider linear correlations, and only correlated the tilt distribution with one other parameter at a time. However, even with these limitations, we find that the current dataset cannot significantly constrain eventual correlations. For all of the models considered in this work, we report Bayesian evidences (Table G.1), which might be used to calculate odd ratios. We find that – within sampling and numerical uncertainties – all of the models are equally supported by the data.
We conclude that the current dataset is not yet large and informative enough to prove that the astrophysical tilt distribution has features, nor that if features exist they manifest as an excess of systems with nearly aligned spin vectors. On the contrary, most of the models we considered yield a broad peak in the astrophysical cosτ distribution at small and positive values. The only conclusion that is consistently found across all models is that there is no excess of systems with negative tilts, relative to what is expected in an isotropic distribution. Our results agree with the literature (e.g., Callister et al. 2022; Tong et al. 2022; Abbott et al. 2021e; Mould et al. 2022) as to the lack of an excess of cosτ ≃ −1, but disagree as to other details (e.g., whether there is a hard cutoff in the cosτ distribution at cosτ < 0 (cfr Callister et al. 2022)). The point of this work is to show that those disagreements are to be expected, given the information in the current dataset. The next observing run of LIGO, Virgo, and KAGRA is scheduled to start in early 2023 (Abbott et al. 2018) and should yield hundreds of BBH sources. Those may yield a first firm measurement of the astrophysical distribution of the tilt angle, and possibly allow us to begin probing correlations with other astrophysical parameters.
Eccentricity is also a powerful indicator of a binary formation channel (Peters 1964; Hinder et al. 2008; Morscher et al. 2015; Samsing 2018; Rodriguez et al. 2018b,a; Gondán & Kocsis 2019; Zevin et al. 2017), but it is currently harder to measure due to the limited sensitivity of groundbased detectors at frequencies below 20 Hz.
For the BBHs detected in the second half of the third observing run (O3b), the LVK has also released tilt posteriors evaluated at minus infinity, i.e., at very large orbital separations (Mould & Gerosa 2022). We ran the Isotropic + Beta model of Sect. 3.2 on O3b sources only, and found that the analyses with tilts calculated at 20 Hz and minus infinity yield the same astrophysical cosτ distribution. We also find that using O3b only sources the cosτ distribution moves toward the left, compared to what is shown in Fig. 6, and peaks closer to 0.
A public repository with the hyperposteriors used in this work is available at https://doi.org/10.5281/zenodo.7305735.
Acknowledgments
The authors would like to thank C. Adamcewicz, V. Baibhav, T. Dent, S. Galaudage, C. Rodriguez and M. Zevin for useful comments and discussion. We would in particular like to thank T. Callister and D. Gerosa for many insightful comments and suggestions. We would like to thank the anonymous A&A Referee, whose comments helped improve the manuscript. S.V. is supported by NSF through the award PHY2045740. S.B. is supported by the NSF Graduate Research Fellowship under Grant No. DGE1122374. CT is supported by the MKI Kavli Fellowship. This material is based upon work supported by NSF’s LIGO Laboratory which is a major facility fully funded by the National Science Foundation. We used publiclyavailable programs BILBY (Ashton et al. 2019; RomeroShaw et al. 2020), DYNESTY (Speagle 2020) and GWPOPULATION (Talbot et al. 2019). This paper carries LIGO document number LIGOP2200275.
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Appendix A: Hierarchical inference
We aim to measure the hyperparameters λ that control the distribution of singleevent parameters θ (the black hole masses, spins, redshifts, etc.) given the dataset D consisting of the 69 GWTC3 BBHs with a false alarm ratio smaller than 1 per year—D ≡ {d_{i}, i = 1…69}—reported by the LVK collaboration (Abbott et al. 2021e). The posterior for λ can be written as follows (Mandel et al. 2019; Fishbach et al. 2018; Vitale et al. 2020):
where we have analytically marginalized over the overall merger rate, which is not relevant for our inference. The function α(λ) represents the detection efficiency, that is to say the fraction of BBHs that are detectable, given the population parameters λ; π(λ) is the prior for the population hyperparameters, and p(d_{i}λ) is the likelihood of the stretch of data containing the ith BBH. This allowed us to account for selection effects and infer the properties of the underlying, rather than the observed, population.
Using Bayes’ theorem and marginalizing over the singleevent parameters, the singleevent likelihood can be written as
where p(θd_{i}, ℋ_{PE}) is the posterior distribution for the binary parameters θ of the ith source. The population hyperparameters λ are typically inferred using a hierarchical process that first involves obtaining posteriors for θ for each individual event under a noninformative prior, π(θℋ_{PE}). The hypothesis ℋ_{PE} represents the settings that were used during this individualevent parameter estimation step. The last term, π(θλ), is the population prior, that is to say our model for how the parameters θ are distributed in the population, given the hyperparameters.
The integral in Eq. A.1 can be approximated as a discrete sum
where the N_{samples} samples are drawn from the posterior distribution of the ith event. We used the posterior samples of the 69 BBHs reported in GWTC3, as released in Abbott et al. (2020b,a, Abbott et al. (2021a,d). For the sources reported in GWTC1, we used the samples labeled IMRPhenomPv2_posterior in the data release; for GWTC2 we used PublicationSamples; for GWTC2.1 we used PrecessingSpinIMRHM; and for GWTC3 we use C01:Mixed. To sample the hyperposterior, we used the DYNESTY (Speagle 2020) sampler available with the GWPopulation package (Talbot et al. 2019).
The detection efficiency α(λ) can also be calculated through an approximated sum starting from a large collection of simulated BBHs for which the signaltonoise ratio (or another detection statistic) is recorded, as described in Farr (2019) and Abbott et al. (2021e). We used the endo3_bbhpopLIGOT2100113v12
123816601815843600.hdf5 sensitivity file released by the LVK (Abbott et al. 2021c) to calculate α(λ), using a false alarm threshold of one per year to identify detectable sources, consistently with Abbott et al. (2021e).
Appendix B: Reference tilt model
Here, we compare our results against the LVK’s model (LVK default) of Abbott et al. (2021e): a mixture between an isotropic component and Gaussian distribution with μ = 1 and an unknown standard deviation,
The Gaussian component was truncated and normalized in the range [ − 1, 1]. The two hyperparameters of LVK default are thus the branching ratio 𝔤of the Gaussian component and its standard deviation σ; they are the same for both black holes. We notice that in Talbot & Thrane (2017), the two normal distributions can assume different values of σ. However, since in general the spins of the least massive objects are measured with extremely large uncertainty, there is no reason to expect that imposing the same distribution on both tilts would introduce biases.
In Fig. B.1 we show the resulting inference on the cosτ, which—modulo differences in sampling settings—is directly comparable to what is presented by the LVK in Abbott et al. (2021e) (their Fig. 15). The colored area shows the 90% CI, the thick dashed line is the median, and the dim lines represent individual draws from the posterior. The two dashed lines represent the edges of the 90% CI obtained by sampling the hyperparameters from their priors. It is worth noticing that the LVK default model excludes, a priori, the possibility of an excess of tilts relative to isotropy (i.e., a posterior larger than 0.5) at negative values, as well as a dearth of tilts relative to isotropy for cosτ ≳ 0.45. Just as Abbott et al. (2021e), we find that the posterior is not entirely inconsistent with a fully isotropic tilt distribution, while preferring an excess of positive alignment.
Fig. B.1. Posterior for cosτ (top panel) and differential merger rate per unit cosτ (bottom panel) obtained using the reference LVK default model. The two thin black dotted lines in the top panel show the 90% CI obtained by drawing the model’s hyperparameters from their priors. In both panels, the thin black lines represent individual posterior draws, whereas the colored band shows the 90% CI. The thick dashed line within the band is the median. 
This is shown in the top panel of Fig. 4, where the solid blue line was obtained using samples from the hyperparameters’ posterior whereas the dashed blue line was obtained by sampling their priors. The fact that there is a hard cutoff at Y = 1 (the finite bin size causes the curves to extend to values slightly smaller than one) is just a symptom of the fact that the LVK default model excludes, a priori, an excess of negative tilts and a dearth at positive tilts, as mentioned above.
The curve is consistent with Y = 1, that is to say isotropic posteriors are perfectly consistent with the data, even though it should be appreciated that the model prefers that region a priori. The level of consistency can also be assessed with Figure 5, which reports, with dashed blue lines, the marginalized posterior on the branching ratio of the isotropic component (as opposed to the Gaussian component, to allow for direct comparisons with other models). While broad, it favors small values for the fraction of sources in the isotropic component, though fully isotropic distributions (𝔦 = 1) are not excluded. The other curves in the figure are discussed in the main body. For all of our models, Tab. G.1 reports the Bayesian evidence, maximum log likelihood, and the number of parameters for the cosτ model, as a differential relative to the default LVK model. That table also includes a fully isotropic model (Isotropic, with p(cosτ_{1}, cosτ_{2})=1/4), which we include as a useful reference. The Isotropic model performs the worst, though not at the point that it can be ruled out with high confidence.
Appendix C: Threecomponent models
The results presented in Sec. 3 show that, depending on the exact model being used, the tilt distribution seems to show either a peak at +1, a peak at a smaller positive value of cosτ, or a broad plateau for positive cosτ. As this might suggest that two peaks, or features, are present in the data, in this Appendix we consider models that are comprised of an isotropic component, plus two other components. To explore the effect of the model on the resulting posterior, we consider different functional forms.
C.1. Isotropic + Gaussian + Beta model
We used a mixture model with an isotropic component, a beta distribution component, and a Gaussian component,
In order to reduce degeneracy between the Gaussian and the beta components, we set the uniform prior for the mean of the Gaussian component to μ ∼ 𝒰(0.9, 5); meanwhile, we restricted the prior of the beta parameters to nonsingular values, α, β ∼ 𝒰(1, 20). In practice, this reduces the possibility that the two components can both create peaks in the same region of the cosτ domain, which would increase degeneracy and hence make sampling more inefficient.
The resulting cosτ posterior is shown in Fig. C.1. The individual posterior draws are colored according to their value of 𝔤; we stress that small (large) 𝔤does not necessarily imply large (small) 𝔟since the isotropic fraction 𝔦 ≡ 1 − 𝔤 − 𝔟 needs not be zero. The 90% CI shows traces of the two features we encountered previously, namely a peak at +1 and one at smaller positive values of cosτ. As with all of the other models explored in this paper (and, to our knowledge, in the literature), we find that the data exclude an excess of black holes with cosτ ≃ −1. The corner plot in Fig. C.2 shows the three branching ratios for this model. The prior for 𝔤and 𝔟was uniform in the plane, with the constraint that 𝔤 + 𝔟 ≤ 1; we show the resulting marginal priors as dotted black lines in the diagonal panels. This model prefers small values of 𝔟coupled with large values of 𝔤, as shown in the topleft offdiagonal panel. There is little posterior support for even moderate values of 𝔟: the 95th percentile for the marginal posterior p(𝔟) is 0.50. As already visible in Fig. C.1, the beta component peaks at small positive values of cosτ: we find .
Fig. C.1. Same as Fig. 7, but for the Isotropic + Gaussian + Beta model. Individual posterior draws are colored according to the branching ratio of the Gaussian component, 𝔤. It is important to note the different scale for the y axis of the bottom panel compared with similar figures for other models. 
Fig. C.2. Joint and marginal posteriors for the branching ratios of all channels for the Isotropic + Gaussian + Beta model. The thin dashed lines in the diagonal plots are the corresponding priors. 
C.2. Isotropic + 2 Gaussians model
Next, we used a mixture model, with an isotropic component, and two Gaussian distributions. Here too, to avoid perfect degeneracy, we somewhat restricted the allowed range of the Gaussian means. The Gaussian on the right (index “R”) has a mean that can only vary in the range [0.9, 5]. The prior for the Gaussian one on the left (index “L”) spans the range [ − 1, 1]; however, we set to 0 the likelihood for samples for which μ_{L} ∉ [A, B], where A and B are hyperparameters of the model. In practice, this implies that the Gaussian on the left is truncated (and hence normalized) in the range [A, B] and has a mean in the same range, for each sample. Mathematically,
We explicitly added hyperparameters for the domain of the left Gaussian in order to verify if the data prefer solutions that do not add posterior support to the antialigned (cosτ ≳ −1) region (cfr. Callister et al. 2022). The resulting posterior for cosτ is shown in Fig. C.3, where the individual posterior draws are colored according to the branching ratio of the right Gaussian – 𝔤_{R}. The 90% CI again shows two features: a rather broad peak for a small positive value of cosτ and a second peak at +1.
Fig. C.3. Same as Fig. 7, but for the Isotropic + 2 Gaussians model, when the left Gaussian component is truncated in the range [A, B], with A and B model’s hyperparameters. Individual posterior draws are colored according to the branching ratio of the rightmost Gaussian component. It is important to note the different scale for the y axis of the bottom panel compared with similar figures for other models. 
We note that the data are informative for the parameters A and B (Fig. C.4). While their priors are uniform, the posteriors for both A and B show clear peaks. For A, we measured , which notably excludes −1 at 90% credibility. Meanwhile, the posterior for B rails against +1. The standard deviation for the left Gaussian, σ_{L}, is large (the 5th percentile of p(σ_{L}d) is 0.46), which implies that even though functionally speaking this component of our model is a Gaussian, the data seem to prefer very wide Gaussians, resembling pieces of segments. The mean of the left Gaussian prefers small positive values, , and shows no obvious correlation with σ_{L}.
Fig. C.4. Joint and marginal posteriors for the hyperparameters and branching ratio associated with the left Gaussian of the Isotropic + 2 Gaussians model. 
Figure C.5 shows the branching ratios for the two Gaussian and the isotropic component 𝔦 ≡ 1 − 𝔤_{R} − 𝔤_{L}, together with the corresponding priors (dashed lines). The measurement is not precise, and only small departures from the priors are apparent. In particular, for both 𝔤_{R} and 𝔤_{L}, the posteriors yield a wide peak at ∼0.5, whereas 𝔦peaks at 0 more than the prior.
Fig. C.6. Same as Fig. C.3, but without truncating the left Gaussian (i.e., with −A = B = 1). Individual posterior draws are colored according to the branching ratio of the rightmost Gaussian component. It is important to note the different scale for the y axis of the bottom panel compared with similar figures for other models. 
Fig. C.8. Joint and marginal posteriors for the hyperparameters and branching ratio of the Tukey component of the Isotropic + Gaussian + Tukey model. 
Fig. C.9. Same as Fig. 7, but for the Isotropic + Gaussian + Tukey model. Individual posterior draws are colored according to the branching ratio of the Gaussian component. It is important to note the different scale for the y axis of the bottom panel compared with similar figures for other models. 
We stress that the posterior on cosτ we obtained for this model is heavily impacted by the fact that the left Gaussian is truncated in a range, whose position is measured from the data. If instead we set −A = B = 1, that is to say if we extend (and normalize) the left Gaussian to the full cosτ range, we would obtain a radically different posterior (Fig. C.6). The branching ratio for the left Gaussian component in this case does not show significant differences relative to the prior. While some of the posterior draws show prominent peaks for small positive values of cosτ, those are not frequent enough to create a visible peak in the 90% CI band, as was instead the case in Fig. C.3.
Given that the only difference between the models behind Fig. C.6 and Fig. C.3 is the truncation of the left Gaussian’s domain, it is tempting to think that the tails of the left Gaussian—if free to extend all the way to cosτ = −1—would give too much posterior weight in that region, which is not supported by the data. This explanation is also consistent with the fact that our model of Sec. C.1 does find the peak, since a beta distribution can produce tails that are less wide than a Gaussian.
C.3. Isotropic + Gaussian + Tukey model
To end our exploration of threecomponent models, we modified the model of the previous section and replaced the left Gaussian with a distribution based on the Tukey window function. Mathematically,
The priors for all of the hyperparameters are uniform, with the exception of the branching ratios 𝔱and 𝔤, which are jointly uniform in the triangle 𝔱 + 𝔤 ≤ 1.
Figure C.7 shows the posteriors for the branching ratios, including that of the isotropic component 𝔦 ≡ 1 − 𝔱 − 𝔤. As for the Isotropic + 2 Gaussians model, the branching ratios are not measured with precision. The data prefer smaller values of 𝔱and 𝔦and moderate values of 𝔤. Comparing Fig. C.8 with the corresponding plot for the Isotropic + Tukey run (Fig. 8), we find qualitatively consistent results. In particular, T_{x0} mostly has support at positive values, except when T_{k} can take large values or when 𝔱is small. Using the full posterior, we find , while restricting to samples with T_{k} ≤ 2 (T_{k} ≤ 1) yields (), which is consistent with the simpler twocomponent model.
Similarly, we find that the posterior for cosτ mainly differs from that of Fig. 7 because of some additional – but not large – support at cosτ = 1, due to the contribution of the Gaussian component.
Appendix D: Correlated mixture models
In this Appendix we revisit some of the models presented in the main body of the paper, and we extend them to allow for the possibility that the hyperparameters governing the populationlevel cosτ distribution are correlated with some of the astrophysical binary parameters. Previous works have considered correlations between the effective aligned spin, χ_{eff}, and the BBH masses and redshifts (Safarzadeh et al. 2020; Callister et al. 2021; Franciolini & Pani 2022; Biscoveanu et al. 2022; Adamcewicz & Thrane 2022), but not a direct correlation between the tilts and these other intrinsic parameters. Given that the number of BBH sources is still relatively small, we only considered a subset of twocomponent models in order to keep the number of hyperparameters limited. For some of the correlated models, the distributions for the two tilt angles are not assumed to be identical (this happens when each tilt is allowed to be correlated with the corresponding component mass or spin magnitude): we only report the distribution for the tilt of a primary (i.e., most massive) black hole, as it is usually best measured.
D.1. Isotropic + correlated Gaussian model
We first allowed for the possibility that the mean and standard deviation of the Gaussian component might be correlated with other astrophysical parameters – κ, described below – since those should be related to the details of the supernovae explosions that would have tilted the orbit (e.g., Gerosa et al. 2018). We minimally modified the Isotropic + Gaussian model to allow the mean and standard deviation to linearly vary with the parameter that is correlated to the spin tilt. This introduces another set of hyperparameters, which control the linear part of the mean and standard deviation:
where and .
It is important to notice that even though we could have also allowed for correlations in the branching ratio 𝔤, we decided not to, as that parameter is already very poorly measured (cfr. Fig 5). Similarly, and following Callister et al. (2021) and Biscoveanu et al. (2021), we only considered linear correlations. As more sources are detected, both of these assumptions might be trivially relaxed. The constant N was chosen to guarantee that the coefficient of μ_{b}, σ_{b} is always smaller than 1. We explored the following possible correlations:
Component masses, κ_{1} = m_{1}, κ_{2} = m_{2}N = 100 M_{⊙}
Component spins, κ_{1} = χ_{1}, κ_{2} = χ_{2}, N = 1
Mass ratio, κ_{1} = κ_{2} = q, N = 1
Total mass, κ_{1} = κ_{2} = m_{tot}, N = 200 M_{⊙}
We found that we cannot constrain, in any significant way, the parameters that enact the correlations (i.e., μ_{b} and σ_{b}), for which we recovered posteriors which highly resemble the corresponding priors. This is shown in Fig. D.1, where we report the parameters of the Gaussian component for the analysis where they are correlated with the component masses. The dashed horizontal lines in the diagonal panels represent the corresponding priors. Especially for the standard deviation term σ_{b}, no information is gained relative to the prior. Because we restricted the prior on σ_{b} to nonnegative values (see Tab. G.2) to ensure that the width of the Gaussian does not become negative for any values of κ, this implies that the overall standard deviation for the Gaussian component can only increase with the mass (Fig. D.2). However, as made clear by comparing the 90% CI band with the extent of the 90% CI obtained with prior draws, the increase of σ is entirely priordriven.
Fig. D.1. Joint and marginal posteriors for the Gaussian parameters obtained in the analysis where they are correlated with the component masses. Dashed lines in the diagonal panels represent the priors. While μ_{a} and σ_{a} resemble the corresponding posterior in the Isotropic + Gaussian model (Fig. 2), the terms that enact the correlation, μ_{b} and σ_{b}, are nearly unconstrained. 
Fig. D.2. Posterior of the standard deviation of the Gaussian component for the Isotropic + correlated Gaussian model, when we allowed for correlation with the component masses, as a function of the primary mass m_{1}. The thin black lines are individual posterior draws, the colored band is the 90% CI, and the thick dashed line is the median. The two horizontal think dotted lines enclose the 90% CI for the Isotropic + Gaussian model, which does not allow for correlations. Finally, the two thin blue dashed lines enclose the 90% CI obtained sampling the prior. 
D.2. Isotropic + correlated Beta model
Fig. D.3. Joint and marginal posteriors for the beta parameters obtained in the analysis where they are correlated with the mass ratio. Dashed lines in the diagonal panels represent the priors. 
Fig. D.4. Same as D.2, but for the β parameter of the Isotropic + correlated Beta model, when correlated with the mass ratio q. 
Fig. D.5. Conditional posteriors for cosτ_{1} for the Isotropic + correlated Beta model, when cosτ is correlated with the mass ratio. Colored bands show the 90% CI posterior for cosτ_{1} conditional on a fixed value of the mass ratio; yellow dashed lines enclose the 90% CI obtained sampling the correlated parameter from its inferred astrophysical distribution; and black dashed lines enclose the 90% CI of the Isotropic + Beta model, which does not allow for correlations. The increased support at +1 as q decreases is mostly priordriven. 
Finally, we augmented the Isotropic + Beta model of Sec. 3.2 to allow for correlations in the parameters that control the beta component:
with and .
We considered the same possible correlations (i.e., values of κ and N) described in the previous section. As for the previous correlated model, we restricted the priors for α_{b} and β_{b} to the nonnegative domain to ensure that the overall α and β parameters of the beta distribution do not become negative, enforcing that only positive correlations can exist between the cosτ distribution and κ. For this model, we found that the current dataset cannot significantly constrain the correlation parameters, even though we did not exactly recover the priors. For example, in Fig. D.3 we show the posterior and priors (thin dashed lines) for the parameters of the beta distribution when we allowed correlations with the mass ratio q. The parameters that enact the correlation, α_{b} and β_{b}, have wide posteriors, which, however, are not as similar as their prior as σ_{b} was for the Isotropic + correlated Gaussian models (Fig. D.1). Figure D.4 shows that the main impact of the measurement, relative to the prior, is to exclude large values of β. However, it is still the case that the overall trend in the 90% CI of the beta parameters are prior dominated. Just as for the Isotropic + correlated Gaussian models, this results in more support at cosτ ≃ +1 for small masses, mass ratios, or spins. Functionally, this happens because the parameters controlling the beta component take smaller values at small values of the correlated parameter, and that moves the peak toward the edge of the domain (for example Fig. D.5 for correlations with the mass ratio). However, just as for the Isotropic + correlated Gaussian model, these trends are mostly a result of the prior and of the model.
Appendix E: Tukey window
We implemented the Tukey window used in Eq. 5 as
This represents a Tukey window that is symmetric around T_{x0} and whose domain is 2T_{k} wide. The parameter T_{r} controls the shape of the window (T_{r} = 0 gives a rectangular window, while T_{r} = 1 gives a cosine). The distribution was then truncated and normalized in the range [ − 1, 1]. Figure E.1 shows four examples. Since the width, the shape, and the position can all be varied, this model is quite elastic and can latch onto both broad and narrow features. We highlight that in the default setting, we allowed the uniform prior of T_{k} to go up to 4 (Tab G.2). This implies that just as for the Isotropic + Gaussian model, there are parts of the parameter space where the nonisotropic component can be made very similar to, or indistinguishable from, the isotropic component. In this case, that happens when T_{k} is large and T_{r} is small. This distribution can also produce curves that ramp up from zero to a plateau, with various degrees of smoothness: the thick green line in Fig. E.1 is an example and—if 𝔱 were zero—would produce a cosτ distribution similar to the second row in Fig. 5 of Callister et al. (2022).
Appendix F: Asymmetry Y(δ) for various values of δ
Appendix G: Tables
Table G.1 reports, for all models (including those discussed later in other appendices), the number of spin parameters, the natural log of the Bayesian evidence, and the natural log of the maximum likelihood point. All quantities are expressed as deltas relative to the reference LVK default model. Table G.2 lists the priors used for the hyperparameters of all models.
Bayesian evidence, maximum log likelihood value, and number of parameters for the cosτ models (relative to the reference LVK model of Eq. B.1).
Priors used for the hyperparameters of the tilt models. All variables are dimensionless.
All Tables
Bayesian evidence, maximum log likelihood value, and number of parameters for the cosτ models (relative to the reference LVK model of Eq. B.1).
Priors used for the hyperparameters of the tilt models. All variables are dimensionless.
All Figures
Fig. 1. Posterior for cosτ (top panel) and differential merger rate per unit cosτ (bottom panel) obtained using the Isotropic + Gaussian model when the mean of the Gaussian component is allowed to vary in the range μ ∈ [ − 1, 1]. The two thin black dotted lines in the top panel show the 90% CI obtained by drawing the model’s hyperparameters from their priors. In both panels, the thin black lines represent individual posterior draws, whereas the colored band shows the 90% CI. The thick dashed line within the band is the median. 

In the text 
Fig. 2. Joint and marginal posteriors for the mean and standard deviation of the Gaussian component for the Isotropic + Gaussian model, as well as for the branching ratio 𝔤, when the mean of the Gaussian component is allowed to vary in the range μ ∈ [ − 1, 1] (dark green) or μ ∈ [ − 5, 5] (light green). 

In the text 
Fig. 3. Same as Fig. 1, but for the Isotropic + Gaussian model, when the mean of the Gaussian component is allowed to vary in the range μ ∈ [ − 5, 5]. 

In the text 
Fig. 4. Unnormalized histogram of Y(δ), the ratio between the probability of cosτ for cosτ ∈ [1 − δ, 1] and cosτ ∈ [ − 1, −1 + δ] for the models of Sects. 3.1–3.3 for δ = 0.01. We also show the result for the LVK default model (Appendix B) for comparison. The solid curves were obtained by sampling the hyperposteriors. The dashed line reports the same quantity, but drawing the model’s hyperparameters from their priors. Values of Y > 1 imply more support for aligned than antialigned spins. 

In the text 
Fig. 5. Marginalized posterior for the branching ratio of the isotropic component – 𝔦 – for all of the uncorrelated twocomponent models (Sects. 3.1–3.3). The figure is split into two panels to enhance clarity. In both panels, we also report the posterior obtained with the LVK default model (yellow dashed line) for comparison. 

In the text 
Fig. 6. Same as Fig. 1, but for the Isotropic + Beta model. It is important to note the different scale for the y axis of the bottom panel compared with similar figures for other models. 

In the text 
Fig. 7. Same as Fig. 1, but for the Isotropic + Tukey model. Individual posterior draws are colored according to the corresponding value of the branching ratio for the Tukey component, 𝔱. 

In the text 
Fig. 8. Joint and marginal posteriors for the hyperparameters and branching ratio of the Tukey component in the Isotropic + Tukey model. 

In the text 
Fig. B.1. Posterior for cosτ (top panel) and differential merger rate per unit cosτ (bottom panel) obtained using the reference LVK default model. The two thin black dotted lines in the top panel show the 90% CI obtained by drawing the model’s hyperparameters from their priors. In both panels, the thin black lines represent individual posterior draws, whereas the colored band shows the 90% CI. The thick dashed line within the band is the median. 

In the text 
Fig. C.1. Same as Fig. 7, but for the Isotropic + Gaussian + Beta model. Individual posterior draws are colored according to the branching ratio of the Gaussian component, 𝔤. It is important to note the different scale for the y axis of the bottom panel compared with similar figures for other models. 

In the text 
Fig. C.2. Joint and marginal posteriors for the branching ratios of all channels for the Isotropic + Gaussian + Beta model. The thin dashed lines in the diagonal plots are the corresponding priors. 

In the text 
Fig. C.3. Same as Fig. 7, but for the Isotropic + 2 Gaussians model, when the left Gaussian component is truncated in the range [A, B], with A and B model’s hyperparameters. Individual posterior draws are colored according to the branching ratio of the rightmost Gaussian component. It is important to note the different scale for the y axis of the bottom panel compared with similar figures for other models. 

In the text 
Fig. C.4. Joint and marginal posteriors for the hyperparameters and branching ratio associated with the left Gaussian of the Isotropic + 2 Gaussians model. 

In the text 
Fig. C.5. Same as Fig. C.2, but for the Isotropic + 2 Gaussians model. 

In the text 
Fig. C.6. Same as Fig. C.3, but without truncating the left Gaussian (i.e., with −A = B = 1). Individual posterior draws are colored according to the branching ratio of the rightmost Gaussian component. It is important to note the different scale for the y axis of the bottom panel compared with similar figures for other models. 

In the text 
Fig. C.7. Same as Fig. C.2, but for the Isotropic + Gaussian + Tukey model. 

In the text 
Fig. C.8. Joint and marginal posteriors for the hyperparameters and branching ratio of the Tukey component of the Isotropic + Gaussian + Tukey model. 

In the text 
Fig. C.9. Same as Fig. 7, but for the Isotropic + Gaussian + Tukey model. Individual posterior draws are colored according to the branching ratio of the Gaussian component. It is important to note the different scale for the y axis of the bottom panel compared with similar figures for other models. 

In the text 
Fig. D.1. Joint and marginal posteriors for the Gaussian parameters obtained in the analysis where they are correlated with the component masses. Dashed lines in the diagonal panels represent the priors. While μ_{a} and σ_{a} resemble the corresponding posterior in the Isotropic + Gaussian model (Fig. 2), the terms that enact the correlation, μ_{b} and σ_{b}, are nearly unconstrained. 

In the text 
Fig. D.2. Posterior of the standard deviation of the Gaussian component for the Isotropic + correlated Gaussian model, when we allowed for correlation with the component masses, as a function of the primary mass m_{1}. The thin black lines are individual posterior draws, the colored band is the 90% CI, and the thick dashed line is the median. The two horizontal think dotted lines enclose the 90% CI for the Isotropic + Gaussian model, which does not allow for correlations. Finally, the two thin blue dashed lines enclose the 90% CI obtained sampling the prior. 

In the text 
Fig. D.3. Joint and marginal posteriors for the beta parameters obtained in the analysis where they are correlated with the mass ratio. Dashed lines in the diagonal panels represent the priors. 

In the text 
Fig. D.4. Same as D.2, but for the β parameter of the Isotropic + correlated Beta model, when correlated with the mass ratio q. 

In the text 
Fig. D.5. Conditional posteriors for cosτ_{1} for the Isotropic + correlated Beta model, when cosτ is correlated with the mass ratio. Colored bands show the 90% CI posterior for cosτ_{1} conditional on a fixed value of the mass ratio; yellow dashed lines enclose the 90% CI obtained sampling the correlated parameter from its inferred astrophysical distribution; and black dashed lines enclose the 90% CI of the Isotropic + Beta model, which does not allow for correlations. The increased support at +1 as q decreases is mostly priordriven. 

In the text 
Fig. E.1. Four examples of the distribution in Eq. E.1. 

In the text 
Fig. F.1. Same as Fig. 4, but for various values of δ. 

In the text 
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