Building better spin models for merging binary black holes: Evidence for non-spinning and rapidly spinning nearly aligned sub-populations

Recent work paints a conflicting portrait of the distribution of black hole spins in merging binaries measured with gravitational waves. Some analyses find that a significant fraction of merging binaries contain at least one black hole with a spin tilt $>90^\circ$ with respect to the orbital angular momentum vector, which has been interpreted as a signature for dynamical assembly. Other analyses find the data are consistent with a bimodal population in which some binaries contain black holes with negligible spin while the rest contain black holes with spin vectors preferentially aligned with the orbital angular momentum vector. In this work, we scrutinize models for the distribution of black hole spins to pinpoint possible failure modes in which the model yields a faulty conclusion. We reanalyze data from the second LIGO--Virgo gravitational-wave transient catalog (GWTC-2) using a revised spin model, which allows for a sub-population of black holes with negligible spins. In agreement with recent results by Roulet et al., we show that the GWTC-2 detections are consistent with two distinct sub-populations. We estimate that $29-75\%$ (90\% credible interval) of merging binaries contain black holes with negligible spin $\chi \approx 0$. The remaining binaries are part of a second sub-population in which the spin vectors are preferentially (but not exactly) aligned to the orbital angular momentum. The black holes in this second sub-population are characterized by spins of $\chi\sim0.45$. We suggest that the inferred spin distribution is consistent with the hypothesis that all merging binaries form via the field formation scenario.


INTRODUCTION
Gravitational waves from merging binaries encode information about the mass and spin of the component black holes and/or neutron stars. These properties, in turn, provides clues as to how the binary formed. Two scenarios are frequently invoked to explain merging binary black holes: field and dynamical (see Mandel & Farmer 2018;Mapelli 2021;Mandel & Broekgaarden 2021 for reviews). In the field scenario, binary black hole (BBH) systems are formed from isolated stellar binaries. In the dynamical scenario, BBH systems are assembled through interactions in dense stellar environ-ments such as globular clusters and nuclear clusters; see Mapelli (2020) for a recent review.
These two scenarios yield distinct predictions for the distribution of black hole spin vectors. Field binaries are generally expected to form BBH systems with dimensionless spin vectors χ 1 , χ 2 preferentially-aligned with the orbital angular momentum vector L. Supernova kicks may serve to somewhat misalign the χ 1,2 and L vectors, but the typical misalignment angle is expected to be modest (O'Shaughnessy et al. 2017;Stevenson et al. 2017;Gerosa et al. 2018;Rodriguez et al. 2016;Bavera et al. 2020). Dynamically assembled binaries on the other hand are expected to form BBH systems with isotropically distributed spin vectors (Kalogera 2000;Mandel & O'Shaughnessy 2010;Rodriguez et al. 2016;Zevin et al. 2017;Talbot & Thrane 2017;Rodriguez et al. 2018;Doctor et al. 2019). These two predictions can be used to estimate the fraction of merging binaries associated with each channel.
In this paper we do not carry out calculations using the second model from Abbott et al. (2021b)-the Gaussian model (Roulet & Zaldarriaga 2019;Miller et al. 2020). However, we argue below that it shares key features with the Default model. We therefore expect that our findings obtained with the Default model are applicable to results obtained with the Gaussian model as well.
Using the Default model, Abbott et al. (2021b) reconstruct the distribution for the effective inspiral spin parameter χ eff ≡ χ 1 cos θ 1 + qχ 2 cos θ 2 1 + q , where q ≡ m 2 /m 1 ≤ 1 is the mass ratio, the distribution of which is modeled according to a power law: The χ eff parameter is frequently used to interpret results because it is an approximate constant of motion in precessing binaries and it is a relatively well-measured quantity. Abbott et al. (2021b) find that 12 − 44% of BBH systems possess χ eff < 0 (90% credible interval), indicating that at least one black hole tilt angle > 90 • . However, this finding is disputed by Roulet et al. (2021). These authors analyze a slightly different set of events (chosen to be more confidently detected) using a distinct set of posterior samples. They carry out two analyses of black hole spin. We focus for the moment on their "model-free" analysis, which assumes that χ eff is uniformly distributed on the interval [−1, 1]. The prior for the poorly-measured variable is chosen to be uniform, conditioned on χ eff and subject to the constraint that χ 1,2 ≤ 1 as required by general relativity. 3 This leaves four degrees of freedom χ ix , χ iy (where i = 1, 2), which are assumed to be uniformly distributed on a disk with radius 1 − χ 2 iz . We call this set of assumptions Roulet+.
Using these distributions, Roulet et al. (2021) plot the credible intervals for χ eff for individual events in their catalog. We reproduce this "dot plot" in Fig. 1 for the events in Abbott et al. (2021b), using posterior samples from that study 4 . Individual event posteriors computed using priors from different population models are represented with different colors: black is the fiducial LIGO-Virgo spin prior (uniform in χ 1 , χ 2 with an isotropic prior for the spin directions), green is the population model from Roulet et al. (2021), and pink is the Default model from Abbott et al. (2021b) (averaged over the posterior on the population parameters).
By inspecting a version of this plot, Roulet et al. (2021) find there is not clear support for a population of events with χ eff < 0. Under all three population models, there are no events which have exclusively negative χ eff support inconsistent with zero. We therefore concur with Roulet et al. (2021) that the BBH mergers observed to date are consistent with a sub-population of negligible-spin events χ 1,2 ≈ 0 with a second subpopulation of events with χ eff > 0. The signature from Abbott et al. (2021b) for a sub-population of χ eff < 0 seen by Abbott et al. (2021b) is likely due to a misspecification of the assumed population spin model. This paper therefore seeks to improve on the Default model by developing a more sophisticated model for black hole spin in order to better describe the data and to provide more reliable inferences.
The remainder of this paper is structured as follows. In Section 2, we discuss why checks in Abbott et al. (2021b), designed to validate the observation of a subpopulation with χ eff < 0, did not reveal the modeldependency of this result. In Section 3, we propose an improved black-hole spin model (called Extended) designed to address the shortcomings of the Default model. We repeat the black hole spin distribution analysis from Abbott et al. (2021b) (Section 4) using the Extended model and report results in Section 5. In agreement with Roulet et al. (2021), we show that the the observation of a sub-population with χ eff < 0 is model-dependent. We present revised fits to the distribution of black hole spins. We conclude in Section 6 by discussing strategies for future development and testing of black hole spin models.

DIAGNOSING LIMITATIONS OF THE
Default MODEL Abbott et al. (2021b) include several checks, which would seem at first glance to support the claim for a sub-population of events with χ eff < 0. First, they note the similar reconstructed distributions for χ eff obtained with two different models: the Default model described above and Gaussian model that describes the distribution of χ eff and the effective precession spin parameter χ p using a multi-variate Gaussian (see their Fig. 11b). The potential for model-induced systematic error is reduced given the signature is visible using two different models. Second, they carry out posterior predictive checks to show that mock source populations drawn from the Gaussian model would have similar distributions of χ eff and χ p to the observed data; see their Fig. 26. Third, they perform an analysis in which a new hyper-parameter χ min eff is added to the population model, which enforces a minimum value of χ eff > χ min eff ; see their Fig. 27. A negative value for χ min eff is preferred over χ min eff = 0, seemingly bolstering the case for a subpopulation of events with χ eff < 0.
It is instructive to discuss how each of these checks fails to catch the apparent lack of χ eff < 0 events in Figure 1.
A "dot plot" showing the medians and 90% credible intervals for χ eff assuming different population models. The fiducial model used by LIGO-Virgo (LVC, black) assumes uniform distribution for dimensionless spins χ1, χ2 with an isotropic distribution for the spin directions. Green (Roulet+) shows the model from Roulet et al. (2021). The Default model from Abbott et al. (2021b) is shown in pink.
No models indicate support for a sub-population of events with χ eff < 0.

Fig. 1. While the agreement between the Default and
Gaussian models provides something of a sanity check, both models lack a key feature: the ability to account for an excess of BBH systems with χ 1,2 ≈ 0. The possibility that some fraction of LIGO-Virgo binaries should merge with negligible spin is supported by theoretical studies of angular momentum transport (Fuller & Ma 2019;Belczynski et al. 2020). Moreover, the negligible spin of many/most LIGO-Virgo binaries has been noted observationally (Miller et al. 2020;Kimball et al. 2021). Neither spin model from Abbott et al. (2021b) accounts for a sub-population of binaries with χ ≈ 0 black holes. Since neither model can accommodate an excess of events with χ 1,2 ≈ 0, the models are liable to fit such an excess with the next best thing: a subpopulation of events distributed about χ eff = 0. In this way, the Default and Gaussian models can both yield false positive signals for χ eff < 0 when the true population contains an excess of events with χ 1,2 ≈ 0.
Next we turn to the posterior predictive check from Abbott et al. (2021b), their Fig. 26. This plot compares the cumulative distribution of χ eff for the data to the distribution expected from the model. The two distributions are visually consistent. While a mismatch between data and model predictions would indicate a failure of the model, a match does not prove that the model is a faithful representation of the data. Using a different population model, Roulet et al. (2021) find no evidence for a sub-population of events with χ eff < 0-a finding that we confirm independently below.
Finally, we consider the test from Abbott et al. (2021b) showing that the data prefer a negative values of χ min eff such that some events in the population are characterized by negative χ eff values on the interval (χ min eff , 0) (see their Fig. 27). It should now be apparent that this test can be tricked if the data contain an excess of events with χ 1,2 ≈ 0. In order to illustrate this-and to build an improved model for BBH spinwe introduce an Extended model, which incorporates elements from Roulet et al. (2021) that enable an excess of negligible-spin events.

THE Extended MODEL
We introduce two changes to the Default model. First, we add a new population parameter λ 0 , corresponding to the fraction of BBH mergers with negligible spin. Our revised spin magnitude distribution is where G t is a truncated Gaussian distribution peaking at χ 1,2 = 0 with width σ 0 . We set σ 0 = 0 so that the spin magnitudes are precisely zero and carry out the analysis using a dedicated set of zero-spin posterior samples. Setting χ = 0 is probably a good approximation since Fuller & Ma (2019) suggest that typical black-hole spin magnitudes are χ ≈ 0.01 (ignoring binary effects, particularly tidally induced spin-up, discussed by, e.g., Kushnir et al. 2016;Zaldarriaga et al. 2018;Qin et al. 2018;Bavera et al. 2020;Belczynski et al. 2020;Mandel & Fragos 2020). It would be interesting to make σ 0 a population parameter that can be fit with the data. However, we leave this for future work as there are some technical challenges; preliminary studies with small, non-zero values of σ 0 seem to suffer from under-sampling effects. We adopt a uniform prior for λ 0 on the interval [0, 1].
Our second change is to add a population parameter z min to the distribution of black-hole tilt angles such that the combined distribution from Eq. 2 is forced to zero for z 1,2 < z min : We adopt a uniform prior for z min on the interval [−1, 1]. In section 5 we present results of an analysis of events from the GWTC-2 catalog using the Extended model. The full list of population hyper-parameters and priors is given in Table 1.

ANALYSIS
We use GWPopulation ) to obtain posterior distributions for the population parameters in the Extended model using the same event list used in Abbott et al. (2021b). This dataset consists of 44 confidently detected BBH mergers. It does not include GW190814 (Abbott et al. 2020b), which may be a neutron-star black-hole binary and which is a clear outlier from the rest of the population. The analysis from Abbott et al. (2021b) employed higher-order modes waveforms, which we are unable to use here for technical reasons. Instead, we use IMRPhenomPv2 and IMR-PhenomD waveforms (Hannam et al. 2014;Husa et al. 2016;Khan et al. 2016) to obtain samples for χ > 0 and χ = 0 respectively. While this choice of waveforms is unlikely to affect our main conclusions, the different choice of waveform leads to subtle shifts in the posterior distribution of some population parameters. We estimate these shifts by comparing the Default model results with IMRPhenomPv2 waveforms to the Default model results with higher-order mode waveforms. We determine that the typical values of χ for the subpopulation with spinning black holes would likely lower Parameter Description Prior λ0 Mixing fraction of mergers with negligible spin, χ1,2 σ0 U(0,1) σ0 Spread in χ1,2 for systems with negligible spin σ0 = 0 µχ Mean of spin magnitude distribution U(0,1) σ 2 χ The square of the width of the spin magnitude distribution U(0,0.25) ζ Mixing fraction of mergers with preferentially aligned spin U(0,1) σt Spread in projected misalignment for preferentially aligned black holes U(0,4) zmin Minimum value of the projected misalignment U(-1,1)  (2018). We fit the merger redshift distribution using the Power-Law Evolution model from Abbott et al. (2021b), adapted from Fishbach et al. (2018). We employ two sets of posterior samples: one using the standard LIGO-Virgo priors and one generated using a zerospin prior; see Kimball et al. (2021). 6 The zero-spin samples are necessary to avoid artifacts due to undersampling. Additional technical details are provided in Appendix A. We adopt the same treatment of selection effects as used in Abbott et al. (2021b), which accounts for massdependent Malmquist bias, but not selection effects due to spin. The authors of that work explain that they cannot reliably estimate spin-induced selection effects using the currently available injection set; see their Appendix F. It is slightly easier to detect BBH signals with χ eff > 0, so we expect that our fit may slightly overemphasize high positive values of χ eff relative to the true distribution. Fig. 1 of Ng et al. (2018) illustrates this shift in the inferred χ eff distribution. This shift will impact the branching ratio between the two non-spinning and spinning sub-populations. We estimate that includ-ing selection effects would decrease the estimated subpopulation of spinning BBHs by 20% of the currently estimated fraction (1−λ 0 ), i.e., 4%. Thus, the systematic error from selection effects is less than the current statistical uncertainty.

RESULTS
We plot population predictive distributions (PPDs) 7 comparing the Default and Extended models in Fig. 2. In the left panel, we show the reconstructed spin magnitude distribution, the Extended model exhibits clear support for a narrow peak at χ 1,2 = 0, which is not present in the Default model because the model lacks the flexibility to fit this peak. In the right panel, we show the reconstructed distribution of the (cosine of the) spin tilts, we see that the Extended model distribution of z 1,2 ≡ cos θ 1,2 tapers off for cos θ 1,2 0 while the Default model exhibits considerably more support for binaries with cos θ 1,2 < 0. Since the Extended model includes the Default model as a special case (with λ 0 = 0, z min = −1), we conclude that the data prefer a sub-population of binaries with χ 1,2 ≈ 0 over a sub-population of binaries with χ eff < 0. This conclusion is supported by our model selection results, summarized in Table 2, which show the Extended model is preferred over the Default model by log 10 B ≈ 3.55. This preference clearly comes from the introduction of a zero spin sub-population (cf. the λ 0 = 0 model). There is a slight preference for the Extended model with a broad uniform prior on z min or with z min fixed to zero than with z min fixed to −1. Figure 3 shows the posterior distributions for the two variables new to the Extended model: z min (left) and λ 0 (right). Turning first to the posterior for z min , we 7 The PPD is given by the conditional prior marginalized over the posterior distribution of the population parameter p Λ (χ 1,2 |d) = dΛ p(Λ|d) π(χ 1,2 |Λ).
Here, Λ are population parameters described in Table 1. find ample posterior support for z min = 0, which means the data are consistent with the premise that all merging binaries have χ eff > 0. Next we turn our attention to the posterior distribution for λ 0 (right), the parameter that controls the fraction of binaries merging with negligible black-hole spin. The distribution shows that majority of BBH systems merge with negligible spins: λ 0 = 0.54 +0.21 −0.25 (90% credibility). In Fig. 4 we include a corner plot showing the posterior distribution for all the population parameters in the Extended model. The navy contours mark the one-, two-, and three-sigma credible intervals for the Extended model while the pink contours show the results for the Default model used in Abbott et al. (2021b). There are a number of interesting differences between the two models. First, the µ χ parameter, which determines the average dimensionless spin magnitude for black holes (those with non-negligible spin in the Extended model), shifts from ≈ 0.3 for Default to ≈ 0.45 for Extended. This shift reflects the fact that dimensionless spin is higher when we allow for a separate population of binaries with negligible black-hole spin. Taken together, Figs. 3 and 4 suggest two subpopulations: one population with BBH systems merging with negligible spins and the other merging with moderately large spins.
Second, the posterior for the ζ parameter (shown in Fig. 4), which determines the fraction of binaries with preferentially aligned spin, becomes broader, approaching a uniform distribution. This change is explained by the introduction of the z min parameter, which provides a new means of building a population of preferentially aligned binaries. 8 Both models disfavor perfect alignment of the black-hole spin vectors with the orbital angular momentum. The Default model shows support for a preferentially aligned population (ζ > 0) but with non-zero misalignment angle spread (σ t > 0). Meanwhile, the Extended model confidently excludes perfect alignment with z min < 0.43 at 99% credibility.
Finally, in Fig. 5, we show the χ eff PPDs for the Default and Extended models. The Extended model is characterized by a sharp peak at χ eff = 0 corresponding to the sub-population of BBH systems with negligible spin. While the Extended PPD does not vanish for χ eff < 0, there is very limited support there, with fewer than 2% of all binaries predicted to have χ eff < −0.1. This is consistent with Fig. 1, which shows no suggestion of events with negative χ eff . This conclusion is also consistent with Fig. 3, which shows support for z min as low as z min ≈ −1. However, the same plot shows that the data are consistent with z min = 0. In other words, we cannot rule out the possibility of a small sub-population with χ eff < 0, but the Extended model provides no evidence that such a sub-population exists. We also performed a set of population predictive checks, which shows the model is a good fit to the data; these figures are available in the companion repository. However, we emphasize that a match of the predicted and observed distributions does not guarantee the model accurately represents the data.

DISCUSSION
Inspired by Roulet et al. (2021), we reanalyze the population of merging binary black holes from Abbott et al. (2021b) using a revised model for the distribution of black hole spins designed to allow for a sub-population with negligible spin. Using the same events and the same posterior samples as Abbott et al. (2021b), we obtain results qualitatively similar to Roulet et al. (2021), suggesting that Abbott et al. (2021b)'s finding of a subpopulation of binaries merging with χ eff < 0 is modeldependent. Using our new Extended model, we find that the data can be explained by the hypothesis that all binaries merge with black hole spin preferentially aligned with the orbital angular momentum or with negligible spin. These findings are consistent with GWTC-1 studies (Farr et al. 2017Abbott et al. 2019;García-Bellido et al. 2021) inferring the population of what was then 10 BBH detections is consistent with mostly negligible spins and a few events with support for χ eff > 0.
This result somewhat diminishes the case for dynamical mergers as a major channel for merging binaries. However, the dynamical scenario remains a plausible explanation for binary mergers with black holes in the pair instability mass gap (Abbott et al. 2020a). If GW190521 is a hierarchical merger (assembled from the products of previous mergers), perhaps black holes participating in  the first generation of dynamical mergers are among the ≈ 54% of with negligible spin. Indeed, studies of dynamical mergers in dense clusters find that small spins are required for first-generation black holes in order for their remnants to be retained to merge again (Kimball et al. 2020(Kimball et al. , 2021. If a large fraction of merging binaries are assembled dynamically, our results suggest that most dynamically merging black holes have negligible spin. A sub-population of binaries with negligible black-hole spins can also be accommodated within the field formation framework; see, e.g., Belczynski et al. (2020);Bavera et al. (2020). If the stellar progenitors of BBH systems efficiently shed angular momentum through mass loss, it may be common for most field BBH systems to form with negligible spin. A minority of progenitors in sufficiently tight binaries could, however, tidally spin up the stellar core of the secondary star prior to its collapse into a black hole, producing a BBH system with at least one rapidly spinning BH. In this framework, black holes with non-negligible spins could be those whose naked helium star progenitors were tidally spun up in tight binaries. This possibility can be explored in the future by relaxing the assumption that the spins and spin-orbit misalignment angles of both binary components are independently drawn from the same distributions.
Binary black hole detections in which the more massive black hole has clearly non-zero spin χ 1 > 0 could be challenging to accommodate in the standard field framework as described above (Mandel & Fragos 2020;Qin et al. 2021). Several events in GWTC-2 would seem to fall into this category, including GW151226  Table 1. The shaded regions represent one-, two-, and three-sigma credible intervals.  Qin et al. 2021, though the high mass already makes field formation unlikely for this system). If these systems formed in the field, it is possible that the more massive black hole formed from what was initially the secondary (lower mass) star that subsequently gained mass through accretion. In this scenario, the secondary star could still be tidally spun up before collapsing into the more massive black hole, a scenario explored in Bavera et al. (2021); Olejak & Belczynski (2021). Furthermore, the progenitor of the first-formed black hole can also be tidally spun up if the binary is initially very compact, as appears to be the case with high-mass Xray binaries such as M33 X-7 (Valsecchi et al. 2010) and Cygnus X-1 (Qin et al. 2019), though it is unclear if such systems lead to merging BBHs (Neijssel et al. 2021).
In this work we focussed only on the black-hole spin distribution, assuming that the black-hole mass distribution is independent. However, there may be possible correlations between mass and spin, as explored by Sa-farzadeh et al. (2020); Tagawa et al. (2021);Callister et al. (2021). Examples include predictions for negative correlation between mass and spin for isolated binary evolution (e.g., Bavera et al. 2020) and mergers in young clusters (e.g., Kumamoto et al. 2021). Meanwhile, positive correlations are predicted for repeated dynamical mergers in globular clusters (e.g., Rodriguez et al. 2018) with a possible high-mass spin-aligned contribution from chemically homogeneously evolving binaries (e.g., Marchant et al. 2016;Mandel & de Mink 2016;Riley et al. 2021), while mergers in AGN disks may exhibit a positive correlation between mass and the dispersion of effective spin (Tagawa et al. 2020). Investigating such correlations, along with possible correlations with redshift, can improve the accuracy of the models and enable them to better distinguish between various evolutionary scenarios.
Our work highlights the subtleties in diagnosing model misspecification, which can lead to overly modeldependent conclusions. While Abbott et al. (2021b) perform a number of checks to validate their evidence for a sub-population of events with χ eff < 0 (see Section 2), none of these checks pinpointed how the model could misbehave when applied to a distribution with a sub-population of binaries with negligible-spin black holes, as pointed out by Roulet et al. (2021). We recommend careful consideration of possible sharp featuresin this case, a narrow peak with χ 1,2 ≈ 0-since they can yield misleading results when fit with slowly varying functions.
The "dot plot" featured in Fig. 1 of Roulet et al. (2021), and included in our own Fig. 1, provides an important visual check. However, it is not obvious that such a plot can be applied more broadly to help spot model misspecification. For one thing, the χ eff dot plot is fairly straightforward to interpret because the likelihood function for individual events is approximately Gaussian in this parameter. However, this is not true for other variables, for example, χ p . Moreover, even the χ eff dot plot includes a hidden population model, which can affect its interpretation.
In order to obtain a prior distribution that is uniform in χ eff , the "model-free" prescription in Roulet et al. (2021) makes implicit assumptions about the distribution of the physical spin parameters χ 1 , χ 2 , cos θ 1 and cos θ 2 . Plotting the distribution of these spin parameters in Fig. 6, we see that some of the assumed distributions are not especially physical. In particular, the "modelfree" analysis implicitly assumes that black holes preferentially spin near the Kerr limit χ 1,2 = 1 with a strong covariance between χ 1 and χ 2 . The distribution of spin tilts, meanwhile, favours spin vectors that are either aligned or anti-aligned with the orbital angular momentum vector. Comparing the green "Roulet +" model to the black LVC model in Fig. 1, we see that the two models produce similar dot plots when one accounts for the implicit preference for larger spins in the "model-free" approach. However, the two models do not produce the same ordered list when we rank events from smallest to largest χ eff , which indicates that other spin degrees of freedom described by the implicit model influence this plot -an effect that will become more significant as the gravitational-wave catalog grows. However, the surprising shape of the distributions in Fig. 6 reminds us that all fully specified models of black-hole spin require some distribution of physical parameters even if these distributions are not explicit. It therefore seems useful to cast population models in terms of physical parameters, or, at the very least, to check the suitability of the distribution of physical parameters.
There are a number of interesting extensions to this work worthy of future exploration. Our Extended model is designed to investigate questions raised in Abbott et al. (2021b) and Roulet et al. (2021). A number of possible model extensions consider more astrophysically motivated distributions. As discussed above, spin magnitudes and directions may be coupled with each other, or with mass and mass ratio (see, e.g., Callister et al. 2021). For example, there could be different spin magnitude distributions in the field scenario (preferentially aligned spins) and the dynamical scenario (isotropic spins). The low-spin-magnitude subpopulation could incorporate small but non-zero spins by varying σ 0 . The z min parameter could be applied only to the sub-population of binaries with cos θ 1,2 described by the truncated Gaussian distribution; this would restore the mixture model choice between a preferentially aligned distribution and an isotropic distribution.
Note: The results in this paper differ from the published version following some fixes to our analysis. A bug in our implementation resulted in the non-spinning posteriors to be given more weight where the fiducial prior per event was 4 times larger than it should have been. The figures and tables in this paper have been updated following this fix.