Distinguishing Spin-Aligned and Isotropic Black Hole Populations With Gravitational Waves

TL;DR

This paper analyzes gravitational wave data from binary black hole mergers to distinguish between spin orientations, finding current evidence slightly favors isotropic distributions, with future data expected to provide stronger confirmation.

Contribution

It introduces a statistical method to differentiate black hole spin distributions using gravitational wave measurements, highlighting the potential to confirm formation scenarios.

Findings

Current data shows a 2.4σ preference for isotropic spins.

Adding more detections increases confidence beyond 5σ.

Future observations will decisively confirm or refute the spin distribution hypothesis.

Abstract

The first direct detections of gravitational waves from merging binary black holes open a unique window into the binary black hole formation environment. One promising environmental signature is the angular distribution of the black hole spins; systems formed through dynamical interactions among already-compact objects are expected to have isotropic spin orientations whereas binaries formed from pairs of stars born together are more likely to have spins preferentially aligned with the binary orbital angular momentum. We consider existing gravitational wave measurements of the binary effective spin, the best-measured combination of spin parameters, in the four likely binary black hole detections GW150914, LVT151012, GW151226, and GW170104. If binary black hole spin magnitudes extend to high values we show that the data exhibit a $2.4\sigma$ ($0.015$ odds ratio) preference for an isotropic angular distribution over an aligned one. By considering the effect of 10 additional detections, we show that such an augmented data set would enable in most cases a preference stronger than $5\sigma$ ($2.9 \times 10^{-7}$ odds ratio). The existing preference for either an isotropic spin distribution or low spin magnitudes for the observed systems will be confirmed (or overturned) confidently in the near future.

Tables

Table 1. Table 1: Significance σ I / A subscript 𝜎 𝐼 𝐴 \sigma_{I/A} of the odds ratio between the isotropic and aligned models using just GW150914 and GW151226, all 3 O1 events, and all 4 currently observed events (including GW170104). The number in bold is the result we quote in the main text.

Events	${\sigma }_{I/A}$	${\sigma }_{I/A}$	${\sigma }_{I/A}$
	“Low”	“Flat”	“High”
GW150914 and GW151226	1.3	2.2	3.7
All O1 events	1.7	2.7	4.4
All O1 events and GW170104	2.4	3.6	5.4

Equations

$${\chi }_{\mathrm{eff}}=\frac{c}{GM}\left(\frac{{\vec{S}}_1}{m_1}+\frac{{\vec{S}}_2}{m_2}\right)\cdot \frac{\vec{L}}{\left|\vec{L}\right|}\equiv \frac{1}{M}\left(m_1{\chi }_1+m_2{\chi }_2\right),$$

$$a_{1,2}=\frac{c}{G{m}_{1,2}^2}\left|{\vec{S}}_{1,2}\right|,$$

$$p(a)=\{\begin{array}{ll}2\left(1-a\right)& \text{“low”}\\ 1& \text{“flat”}\\ 2a& \text{“high”}\end{array},$$

$$p\left(\left\{{\chi }_{\mathrm{eff}}^i\right\},\lambda \mid \left\{d^i\right\}\right)\propto \left[\prod _{i=1}^{N_{\mathrm{obs}}}p\left(d^i\mid {\chi }_{\mathrm{eff}}^i\right)p\left({\chi }_{\mathrm{eff}}^i\mid \lambda \right)\right]p\left(\lambda \right).$$

$$p\left(\lambda \mid \left\{d^i\right\}\right)\propto \left[\prod _{i=1}^{N_{\mathrm{obs}}}\int \mathrm{d}{\chi }_{\mathrm{eff}}^{i}p\left(d^i\mid {\chi }_{\mathrm{eff}}^i\right)p\left({\chi }_{\mathrm{eff}}^i\mid \lambda \right)\right]p\left(\lambda \right).$$

$$p\left(\lambda \mid \left\{d^i\right\}\right)\propto \left[\prod _{i=1}^{N_{\mathrm{obs}}}\frac{1}{N_i}\sum _{j=1}^{N_i}\frac{p\left({\chi }_{\mathrm{eff}}^{ij}\mid \lambda \right)}{p\left({\chi }_{\mathrm{eff}}^{ij}\right)}\right]p\left(\lambda \right).$$

$$\frac{p\left(d\mid I\right)}{p\left(d\mid A\right)}\simeq \frac{P\left(-0.25\le {\chi }_{\mathrm{eff}}\le 0.25\right)}{P\left(0\le {\chi }_{\mathrm{eff}}\le 0.5\right)},$$

$$p(a)\propto (1-a{)}^{\alpha }$$

Full text

{affiliations}

Birmingham Institute for Gravitational Wave Astronomy and School of Physics and Astronomy, University of Birmingham, Birmingham, B15 2TT, United Kingdom

Department of Astronomy and Joint Space-Science Institute, University of Maryland, College Park, MD 20742–2421, United States

Kavli Institute for Theoretical Physics, Santa Barbara, CA 93106

Enrico Fermi Institute and Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL 60637, United States

Following the detection of a merging binary black hole system, parameter estimation tools compare model gravitational waveforms against the observed data to obtain a posterior distribution on the parameters that describe the compact binary source. The spin parameter with the largest effect on waveforms, and a correspondingly tight constraint from the data[3], is a mass-weighted combination of the components of the dimensionless spin vectors of the two black holes that are aligned with the orbital axis, the “effective spin,” $-1<\chi_{\mathrm{eff}}<1$ (see Methods Section 2).

Figure 1 shows an approximation to the posterior inferred on $\chi_{\mathrm{eff}}$ for the four likely gravitational wave (GW) detections GW150914, GW151226, GW170104, and LVT151012 from Advanced LIGO’s first and second observing runs (O1 and O2)[3, 4]. Because samples drawn from the posterior on $\chi_{\mathrm{eff}}$ are not publicly released at this time, we have approximated the posterior as a Gaussian distribution with the same mean and 90% credible interval, truncated to $-1<\chi_{\mathrm{eff}}<1$. None of the $\chi_{\mathrm{eff}}$ posteriors are consistent with two black holes with large aligned spins, $\chi_{1,2}\gtrsim 0.5$; this contrasts with the large spins inferred for the majority of black holes in X-ray binaries with claimed spin measurements[16] (see below). The analysis here is relatively insensitive to the precise details of the posterior distributions; other conclusions are more sensitive. In particular, our Gaussian approximation does permit $\chi_{\mathrm{eff}}=0$ for GW151226 while the true posterior rules this out at high confidence[2, 3].

Small values of $\chi_{\mathrm{eff}}$ as exhibited in these systems can result from either intrinsically small spins or larger spins whose direction is mis-aligned with the orbital angular momentum of the binary (i.e. spin vectors with small $z$-components). Mis-alignment is capable of producing negative values of $\chi_{\mathrm{eff}}$, however, whereas aligned spins will always have $\chi_{\mathrm{eff}}\geq 0$. This difference provides strong discriminating power between the two angular distributions, even without good information about the magnitude distribution; to the extent that data favour negative $\chi_{\mathrm{eff}}$ they weigh heavily against aligned models. To quantify the degree of support for these two alternate explanations of small $\chi_{\mathrm{eff}}$ values in the merging binary black hole population, we compared the Bayesian evidence for various simple models of the spin population using the GW data set.

Each of our models for the merging binary black hole spin population assumes that the merging black holes are of equal mass (this is marginally consistent with the observations[3, 4], and the $\chi_{\mathrm{eff}}$ distribution is not sensitive to the mass ratio—see Methods Section 6). We assume that the population spin distribution factorises into a distribution for the spin magnitude $a$ and a distribution for the spin angles. Finally, we assume that the distribution of spins is common to each component in a merging binary (the distributions of spin for each component in the binary could differ systematically due to different formation histories). Choosing one of three magnitude distributions (see Methods Section 2), “low” (mean $a=0.33$, standard deviation $0.24$), “flat” (mean $a=0.5$, standard deviation $0.29$), “high” (mean $a=0.67$, standard deviation $0.24$) and pairing with an isotropic angular distribution or a distribution that generates perfect alignment yields six different models for the $\chi_{\mathrm{eff}}$ distribution. These models are shown in Figure 2.

These magnitude distributions are not meant to represent any particular physical model, but rather to capture our uncertainty about the spin magnitude distribution; neither observations nor population synthesis codes can at this point authoritatively suggest any particular spin distribution[16]. Our models, however, allow us to see how sensitive the $\chi_{\mathrm{eff}}$ distribution is to spin alignment given uncertainties about the spin magnitudes.

We fit hierarchical models of the three existing LIGO O1 and GW170104 observations using these six different, zero-parameter population distributions (see Methods Section 4). We also fit three mixture models for the population, where the angular distribution is a weighted sum of the isotropic and aligned distributions. The evidence, or marginal likelihood, for each of the models is shown in Figure 3. For all three magnitude distributions, the mixture models’ posterior on the mixing fraction peaks at 100% isotropic. Not surprisingly, given the small $\chi_{\mathrm{eff}}$ values in the three detected systems, the most-favoured model among those with an isotropic angular distribution has the “low” magnitude distribution; the most favoured model among those with an aligned distribution also has the “low” magnitude distribution. The odds ratio between the “low” aligned and “low” isotropic models is $0.015$, or $2.4\sigma$; thus the data favour isotropic spins among our suite of models. While the data favour spin amplitude distributions with small spin magnitudes, note that a model with all binary black hole systems having zero spin is ruled out by the GW151226 measurements, which bound at least one black hole to have spin magnitude $\geq 0.2$ at 99% credibility[2].

Estimates of the rate of binary black hole coalescences give a reasonable chance of 10 additional binary black hole detections in the next three years[3, 15]. Assuming 10 additional detections drawn from each of our six zero-parameter models in addition to the four existing detections from O1 and GW170104, with observational uncertainties drawn randomly from the three Gaussian widths used to approximate the $\chi_{\mathrm{eff}}$ posteriors in Figure 1222The measurement uncertainty in $\chi_{\mathrm{eff}}$ depends on the other parameters of the merging binary black hole system, particularly on the mass ratio. Our assumption about future observational uncertainties is appropriate if the parameters of the three detected events are representative of the parameters of future detections. See Methods Section 8 for further discussion., we find the odds ratios shown in Figure 4. We find that most scenarios with an additional 10 detections allow the simulated angular distribution to be inferred with greater than $5\sigma$ ($2.9\times 10^{-7}$ odds) credibility. In the most pessimistic case the distinction is typically $2.4\sigma$ ($0.016$ odds ratio). While such future detections should permit a confident distinction between angular distributions, we would remain much less certain about the magnitude distribution among the three options considered here until we have a larger number of observations.

Most of our resolving power for the spin angular distribution is a result of the fact that our “aligned” models cannot produce $\chi_{\mathrm{eff}}<0$ (see Figure 2). If spins are intrinsically very small, with $a\lesssim 0.2$, then it is no longer possible to resolve the negative effective spin with a small number of observations (see Methods Section 5). As noted below, however, spins observed in X-ray binaries are typically large. Additionally, models which do not permit some spins with $\chi_{\mathrm{eff}}\gtrsim 0.1$ are ruled out by the GW151226 observations[2]. An “aligned” model with spin magnitudes from our “flat” distribution but permitting spin vectors oriented anti-parallel to the orbital angular momentum (leading to the possibility of positive or negative $\chi_{\mathrm{eff}}$) can only be distinguished from an isotropic true population at $\sim 3\sigma$ after 10–20 observations[17]; our flat aligned model can be distinguished from such a population at better than $5\sigma$ (odds $<10^{-8}$) after 10 observations, emphasizing the information content of the bound $\chi_{\mathrm{eff}}>0$ for our aligned models.

Observational data on spin magnitudes in black hole systems is sparse[16]. Most of the systems studied are low-mass X-ray binaries rather than the high-mass X-ray binaries that are likely to be the progenitors of double black hole binaries. In addition, there are substantial systematic errors that can complicate these analyses[16] and selection effects could yield a biased distribution. Nonetheless, if we take the reported spin magnitudes as representative then we find that there is a preference for high spins; for example, 14 of the 19 systems with reported spins have dimensionless spin parameters in excess of 0.5. It is usually argued that the masses and spin parameters of stellar-mass black holes are unlikely to be altered significantly by accretion[18], but this may not be true for all systems[19]. Thus the current spin parameters are probably close to their values upon core collapse, at least in high-mass X-ray binaries. However, the specific processes involved in the production of black hole binaries from isolated binaries could alter the spin magnitude distribution of those holes relative to the X-ray binary systems; for example, close tidal interactions could spin up the core, or stripping of the envelope could reduce the available angular momentum[20, 21, 22].

The spin directions in isolated binary black holes[10, 11, 12, 13, 14] are usually expected to be preferentially aligned. Despite observed spin-orbit misalignments in massive stellar binaries[23], mass transfer and tidal interactions will tend to realign the binary. On the other hand, there is some evidence of spin-orbit misalignment in black hole X-ray binaries[24, 25]. This is consistent with the expectation that a supernova natal kick (if any) can change the orbital plane and misalign the binary[26]; the supernova can also tilt the spin angle[27]. Evolutionary processes, such as wind-driven mass loss and post-collapse fallback, can couple the spin magnitude and direction distributions, contrary to our simplified assumptions. A small misalignment at wide separation can also evolve to a more significant misalignment in component spins as the binary spirals in through GW emission[28], but $\chi_{\mathrm{eff}}$ is approximately conserved through this evolution.

The spin directions of binary black holes formed dynamically through interactions in dense stellar environments[5, 6, 7, 8] are expected to be isotropic given the absence of a preferred direction[9] and the persistence of an isotropic distribution through post-Newtonian evolution[29, 30].

{addendum}

We thank Richard O’Shaughnessy, Christopher Berry, Davide Gerosa, and Salvatore Vitale for discussions and comments on this work. WF, SS, IM and AV were supported in part by the STFC. MCM acknowledges support of the University of Birmingham Institute for Advanced Study Distinguished Visiting Fellows program. SS and IM acknowledge support from the National Science Foundation under Grant No. NSF PHY11-25915.

Supplementary Information is linked to the online version of the paper at http://www.nature.com/nature.

All authors contributed to the work presented in this paper.

Reprints and permissions information is available at http://www.nature.com/reprints. The authors declare no competing financial interests. Correspondence and requests for materials should be addressed to w.farr@bham.ac.uk.

{methods}

1 Code Availability

This analysis used the Julia language[31], Python libraries NumPy and SciPy[32, 33], the plotting library Matplotlib[34], and performed computations in IPython notebooks[35]. A repository containing the code and notebooks used for this analysis, together with the LaTeX source for this document, can be found under an open-source “MIT” license at https://github.com/farr/AlignedVersusIsoSpin.

2 Effective Spin and Spin Magnitude Distributions

The effective spin is defined by[36]

[TABLE]

where $m_{1,2}$ are the gravitational masses of the more-massive (1) and less-massive (2) components, $M=m_{1}+m_{2}$ is the total mass, $\vec{S}_{1,2}$ are the spin angular momentum vectors of the black holes in the binary, $\vec{L}$ is the orbital angular momentum vector, assumed to point in the $\hat{z}$ direction, and $\chi_{1,2}$ are the corresponding dimensionless projections of the individual BH spins. Because the dimensionless spin parameter,

[TABLE]

of each black hole is bounded by $0\leq a_{1,2}<1$, the projections along the orbital axis are bounded by $-1<\chi_{1,2}<1$, and $-1<\chi_{\mathrm{eff}}<1$.

We form the population distributions of $\chi_{\mathrm{eff}}$ shown in Figure 2 by assuming that each black hole in a binary has a dimensionless spin magnitude drawn from one of three distributions,

[TABLE]

referred to as “low,” “flat,” and “high” in the text above. These distributions are shown in Extended Data Figure 1.

3 Mixture model

While we carried out Bayesian comparisons between isotropic and aligned spin distributions under various assumptions, a preference for one of the considered models over the others does not necessarily indicate that it is the correct model. All of the considered models could be inaccurate for the actual distribution, especially since all of the considered models are based on a number of additional assumptions, such as decoupled spin magnitude and spin misalignment angle distributions and identical distributions for primary and secondary spins.

We now partly relax the simplified assumptions made earlier by considering the possibility that the true distribution of BBH spin-orbit misalignments observed by LIGO is a mixture of binaries with aligned spins and binaries with isotropic spins.

We fit a mixture model[37] (labelled model ’M’ in Figure 3) where a fraction $f_{i}$ of BBHs have spins drawn from an isotropic distribution, whilst a fraction $1-f_{i}$ have their spins aligned with the orbital angular momentum. We assume a flat prior on the fraction $f_{i}$. To test the robustness of our result, we vary the distribution we assume for BH spin magnitude distributions as with the aligned and isotropic models. We use the “flat”, “high” and “low” distributions (Equation 3), assuming all BHs have their spin magnitude drawn from the same distribution for both the aligned and isotropic populations. We calculate and plot the posterior on $f_{i}$ given by Equation 6 ($f_{i}=\lambda$ in the derivation) in Extended Data Figure 2. We find the mean fraction of BBHs coming from an isotropic distribution is 0.70, 0.77 and 0.81 assuming the “low”, “flat” and “high” distributions for spin magnitudes respectively, compared to the prior mean of 0.5. The lower 90% limits are 0.38, 0.51 and 0.60 respectively, compared to the prior of 0.1. In all cases, the posterior peaks at $f_{i}=1$. Thus, for these spin magnitude distributions we find that the current O1 and GW170104 LIGO observations constrain the majority of BBHs to have their spins drawn from an isotropic distribution. The evidence ratios of these mixture models to the isotropic distribution with “low” spin magnitudes are 0.43, 0.20 and 0.10 for the “low”, “flat” and “high” spin magnitude models. Thus we cannot rule out a mixture with the current data. If several different components contribute significantly to the true spin distribution it may take tens to hundreds of detections to accurately determined the mixing fraction, depending on the distribution of spin magnitudes[17, 37].

4 Hierarchical Modelling

LIGO measures $\chi_{\mathrm{eff}}$ better than any other spin parameter, but still with significant uncertainty[38], so we need to properly incorporate measurement uncertainty in our analysis; thus our analysis must be hierarchical[39, 40]. In a hierarchical analysis, we assume that each event has a true, but unknown, value of the effective spin, drawn from the population distribution, which may have some parameters $\lambda$; then the system is observed, represented by the likelihood function, which results in a distribution for the true effective spin (and all other parameters describing the system) consistent with the data. Combining, the joint posterior on each system’s $\chi_{\mathrm{eff}}^{i}$ parameters and the population parameters $\lambda$ implied by a set of observations each with data $d^{i}$, is

[TABLE]

The components of this formula are

• •

The GW (marginal) likelihood, $p\left(d\mid\chi_{\mathrm{eff}}\right)$. Here we use “marginal” because we are (implicitly) integrating over all parameters of the signal but $\chi_{\mathrm{eff}}$. Note that it is the likelihood rather than the posterior that matters for the hierarchical analysis; if we are given posterior distributions or posterior samples, we need to re-weight to “remove” the prior and obtain the likelihood.

• •

The population distribution for $\chi_{\mathrm{eff}}$, $p\left(\chi_{\mathrm{eff}}\mid\lambda\right)$. This function can be parameterised by population-level parameters, $\lambda$. (In the cases discussed above, there are no parameters for the population.)

• •

The prior on the population-level parameters, $p(\lambda)$.

If we do not care about the individual event $\chi_{\mathrm{eff}}$ parameters, we can integrate them out, obtaining

[TABLE]

If we are given posterior samples of $\chi_{\mathrm{eff}}^{ij}$ ($i$ labels the event, $j$ labels the particular posterior sample) drawn from an analysis using a prior $p\left(\chi_{\mathrm{eff}}\right)$, then we can approximate the integral by a re-weighted average of the population distribution over the samples (here $p\left(\chi_{\mathrm{eff}}^{ij}\right)$ is the prior used to produce the posterior samples):

[TABLE]

4.1 Order of Magnitude Calculation

It is possible to estimate at an order-of-magnitude level the rate at which evidence accumulates in favour of or against the isotropic models as more systems are detected. Based on Figure 2, approximate the isotropic population $\chi_{\mathrm{eff}}$ distribution as uniform on $\chi_{\mathrm{eff}}\in\left[-0.25,0.25\right]$ and the aligned population $\chi_{\mathrm{eff}}$ distribution as uniform on $\chi_{\mathrm{eff}}\in\left[0,0.5\right]$. Then the odds ratio between the isotropic and aligned models for each event is approximately

[TABLE]

where $P\left(A\leq\chi_{\mathrm{eff}}\leq B\right)$ is the posterior probability (here used to approximate the likelihood) that $\chi_{\mathrm{eff}}$ is between $A$ and $B$. Using our approximations to the $\chi_{\mathrm{eff}}$ posteriors described above, this gives an odds ratio of $5$ in favour of the isotropic models, which is about a factor of two smaller than the ratio in the more careful calculation described above. This is a satisfactory answer at an order-of-magnitude level.

If the true distribution is isotropic and follows this simple model, and our measurement uncertainties on $\chi_{\mathrm{eff}}$ are $\simeq 0.1$, then the geometric mean of each subsequent measurement’s contribution to the overall odds is $\sim 3$. After ten additional events, then, the odds ratio becomes $5\times 3^{10}\simeq 3\times 10^{5}$, or $4.6\sigma$, consistent with the results of the more detailed calculation described above. If the true distribution of spins becomes half as wide ($\chi_{\mathrm{eff}}\in[-0.125,0.125]$ for isotropic and $\chi_{\mathrm{eff}}\in[0,0.25]$ for aligned spins), with the same uncertainties, then the existing odds ratio becomes $1.08$, and each subsequent event drawn from the isotropic distribution contributes on average a factor of $1.6$. In this case, after 10 additional events, the odds ratio becomes $150$, or $2.7\sigma$. With small spin magnitudes, our angular resolving power vanishes, as discussed in more detail in Methods Section 5.

4.2 Accumulation of evidence

In Table 1 we show how the evidence for an isotropic distribution increases when including: only the 2 confirmed events—GW150914 and GW151226—from O1; all O1 events (including LVT151012); and all 4 likely binary black hole mergers, including GW170104.

5 Effect of small spin magnitudes

In the main text we considered three models for BH spin magnitudes: “low”, “flat” and “high”. These were intended to capture some of the uncertainty regarding the BH spin magnitude distribution. We may remain observationally uncertain about the spin magnitude distribution until we have $\mathcal{O}(100)$ observations[41, 42].

Here we extend the “low” model as:

[TABLE]

When $\alpha=0$, this recovers the “flat” distribution, whilst $\alpha=1$ recovers the “low” distribution. For higher values of $\alpha$, this distribution becomes more peaked towards $a=0$.

In Extended Data Figure 3 we plot the evidence ratio of isotropic to aligned distributions (plotted as the equivalent sigma) with spin magnitudes given by this model with $\alpha$ in the range [math]–$6$. The top axis shows the mean spin magnitude that value of $\alpha$ corresponds to (e.g., for the “flat” distribution $\alpha=0$, the mean spin magnitude is 0.5). We see that if typical BH spins are $\lesssim 0.2$ we have no evidence for one model over the other.

6 Mass Ratio

Extended Data Figure 4 shows the distributions of $\chi_{\mathrm{eff}}$ that would obtain with a mass ratio $q=m_{2}/m_{1}=0.5$ compared to the distributions with $q=1$ used above. The details of the distribution are sensitive to the mass ratio, but in our analysis we are primarily sensitive to the changing sign of $\chi_{\mathrm{eff}}$ under the isotropic models. This latter property is insensitive to mass ratio. As an example, the distinction between the three different spin amplitude distributions after ten additional detections is quite weak compared to the aligned/isotropic distinction in Figure 4. The differences in the $\chi_{\mathrm{eff}}$ distribution between $q=1$ and $q=0.5$ are even smaller than the differences between the different magnitude distributions.

7 Approximations in the Gravitational Waveform and Selection Effects

While the Advanced LIGO searches use spin-aligned templates they are efficient in detecting misaligned binary black hole systems[43]; we assume here that the $\chi_{\mathrm{eff}}$ distribution of observed sources follows the true population.

The model waveforms used to infer the $\chi_{\mathrm{eff}}$ of the three LIGO events incorporate approximations to the true behaviour of the merging systems that are expected to break down for sufficiently high mis-aligned spins. The effect of these approximations on inference on the parameters describing GW150914 has been investigated in detail[44]. For this source, statistical uncertainties dominate over any waveform systematics. Detailed comparisons with numerical relativity computations using no approximations to the dynamics[45] also suggest that statistical uncertainties dominate the systematics for this system. Systematics may dominate for signals with this large SNR ($\simeq 23$) when the source is edge-on or has high spins[44]. The other two events discussed in this paper are at much lower SNR, with correspondingly larger statistical uncertainties, and are probably similarly oriented and with similarly small spins, so we do not expect systematic uncertainties to dominate.

We assume here that measurements made in the future are not dominated by systematic errors, but this assumption would need to be revisited for high-SNR, edge-on, or high-spin sources detected in the future.

8 Precision of $\chi_{\mathrm{eff}}$ measurements

Throughout this work we have made the simplifying assumption that the precision to which $\chi_{\mathrm{eff}}$ can be constrained for individual binaries is independent of the binary’s properties. In practice, our ability to constrain $\chi_{\mathrm{eff}}$ is dependent on the system’s properties, in particular its true $\chi_{\mathrm{eff}}$ and mass ratio, which we illustrate in Extended Data Figure 5.

For this figure a detected population333We qualify a system as “detected” if it produces a SNR above $8$ in the second-loudest detector to select only coincident events. of 500 binaries was selected from a population with component masses distributed uniformly between $1$ and $30~{}\mathrm{M}_{\odot}$ with $m_{1}+m_{2}<30~{}\mathrm{M}_{\odot}$, locations distributed uniformly in volume, and orientations distributed isotropically. Data were simulated for each binary, and posteriors were estimated using the LIGO-Virgo parameter estimation library LALInference[38] using inspiral-only waveform models (merger and ringdown effects can provide additional information for some binaries, but we ignore those effects here). $\chi_{\mathrm{eff}}$ is better constrained for binaries with high effective spins and high ($\sim$equal) mass ratios.

We do not expect these effects to qualitatively affect out conclusions, though they could affect predictions for the total number of detections necessary to constrain the population. For example, if the universe preferentially forms asymmetric binaries with low mass ratios, individual $\chi_{\mathrm{eff}}$ constraints will be systematically worse, requiring more binaries to infer the properties of the population.