The direct detection of gravitational waves1,2,3,4 from merging binary black holes opens up a window into the environments in which binary black holes form. One signature of such environments is the angular distribution of the black hole spins. Binary systems that formed through dynamical interactions between already-compact objects are expected to have isotropic spin orientations5,6,7,8,9 (that is, the spins of the black holes are randomly oriented with respect to the orbit of the binary system), whereas those that formed from pairs of stars born together are more likely to have spins that are preferentially aligned with the orbit10,11,12,13,14. The best-measured combination of spin parameters3,4 for each of the four likely binary black hole detections GW150914, LVT151012, GW151226 and GW170104 is the ‘effective’ spin. Here we report that, if the magnitudes of the black hole spins are allowed to extend to high values, the effective spins for these systems indicate a 0.015 odds ratio against an aligned angular distribution compared to an isotropic one. When considering the effect of ten additional detections15, this odds ratio decreases to 2.9 × 10−7 against alignment. 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.
After 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 data3, 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, referred to as the ‘effective spin’ −1 < χeff < 1 (see Methods section ‘Distributions of effective spin and spin magnitude’).
In Fig. 1 we show an approximation to the posterior inferred on χeff for the four likely gravitational wave 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 χeff 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 < χeff < 1. None of the χeff posteriors is consistent with two black holes with large aligned spins, χ1,2⋧0.5; this contrasts with the large spins that are inferred for the majority of black holes in X-ray binaries with spin measurements16 (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 χeff = 0 for GW151226, whereas the true posterior rules this out at high confidence2,3.
Small values of χeff as exhibited in these systems can result from either intrinsically small spins or larger spins whose direction is misaligned with the orbital angular momentum of the binary (that is, spin vectors with small z components). However, misalignment can produce negative values of χeff, whereas aligned spins will always have χeff ≥ 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 χeff, they weigh heavily against aligned models independently of assumptions about the spin magnitude distribution. To quantify the degree of support for these two alternative explanations of small χeff values in the merging binary black hole population, we compared the Bayesian evidence for various simple models of the spin population using the gravitational wave dataset.
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 observations3,4, and the χeff distribution is not sensitive to the mass ratio; see Methods section ‘Mass ratio’). We assume that the spin distribution of the population factorizes 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 owing to different formation histories). Choosing one of three magnitude distributions—‘low’ (mean, a = 0.33; standard deviation, 0.24), ‘flat’ (mean, a = 0.5; standard deviation, 0.29) or ‘high’ (mean, a = 0.67; standard deviation, 0.24); see Methods section ‘Distributions of effective spin and spin magnitude’—and pairing with an isotropic angular distribution or a distribution that generates perfect alignment yields six different models for the χeff distribution. These models are shown in Fig. 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 distribution16. However, our models allow us to determine how sensitive the χeff distribution is to spin alignment given uncertainties about the spin magnitudes.
We fit hierarchical models of the four existing observations (three from LIGO’s O1 plus GW170104) using these six different, zero-parameter population distributions (see Methods section ‘Hierarchical modelling’). We also fit three mixture models for the population, in which 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 Fig. 3. For all three magnitude distributions, the posterior on the mixing fraction from the mixture models peaks at 100% isotropic. Not surprisingly, given the small χeff 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 2.4σ; the data thus favour isotropic spins among our suite of models. Although the data favour spin amplitude distributions with small spin magnitudes, a model in which all binary black hole systems have zero spin is ruled out by the GW151226 measurements, which bound at least one black hole to have a spin magnitude of ≥0.2 at 99% credibility2.
Estimates of the rate of binary black hole coalescences give a reasonable chance of 10 additional binary black hole detections in the next three years3,15. Assuming 10 additional detections drawn from each of our six zero-parameter models in addition to the four existing detections from LIGO’s O1 and GW170104, with observational uncertainties drawn randomly from the three Gaussian widths used to approximate the χeff posteriors in Fig. 1, we find the odds ratios shown in Fig. 4. The measurement uncertainty in χeff 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 ‘Precision of χeff measurements’ for further discussion. We find that most scenarios with an additional 10 detections allow the simulated angular distribution to be inferred with greater than 5σ credibility (2.9 × 10−7 odds ratio). In the most pessimistic case, the distinction is typically 2.4σ (0.016 odds ratio). Although 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 χeff < 0 (see Fig. 2). If spins are intrinsically very small, with a≲0.2, then it is no longer possible to resolve the negative effective spin with a small number of observations (see Methods section ‘Effect of small spin magnitudes’). As noted below, however, spins observed in X-ray binaries are typically large. Additionally, models that do not permit some spins with χeff⋧0.1 are ruled out by the GW151226 observations2. An aligned model with spin magnitudes from our flat distribution but permitting spin vectors oriented antiparallel to the orbital angular momentum (leading to the possibility of positive or negative χeff) can be distinguished from an isotropic true population at about 3σ only after 10–20 observations17. In contrast, our flat, aligned model can be distinguished from such a population at better than 5σ (<10−8 odds ratio) after 10 observations, emphasizing the information content of the bound χeff > 0 for our aligned models.
Observational data on spin magnitudes in black hole systems is sparse16. 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 analyses16 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 greatly altered by accretion18, but this may not be true for all systems19. The current spin parameters are therefore probably close to their values upon core collapse, at least in high-mass X-ray binaries. However, the specific processes that are involved in the production of black hole binaries from isolated binaries could alter the spin magnitude distribution of those black holes relative to the X-ray binary systems; for example, close tidal interactions could increase the spin magnitude of the core, and stripping of the envelope could reduce the available angular momentum20,21,22.
The spin directions in isolated binary black holes10,11,12,13,14 are usually expected to be preferentially aligned. Although spin–orbit misalignments have been observed in massive stellar binaries23, mass transfer and tidal interactions tend to realign the binary. On the other hand, there is some evidence of spin–orbit misalignment in black hole X-ray binaries24,25. This evidence is consistent with the expectation that a supernova natal kick could change the orbital plane and misalign the binary26; the supernova could also tilt the spin angle27. 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 substantial misalignment in component spins as the binary spirals in, as a result of the emission of gravitational waves28, but χeff is approximately conserved throughout this evolution.
The spin directions of binary black holes that are formed dynamically through interactions in dense stellar environments5,6,7,8 are expected to be isotropic, owing to the absence of a preferred direction9 and the persistence of an isotropic distribution throughout in-spiralling29,30.
Distributions of effective spin and spin magnitude
The effective spin is defined by31where m1,2 are the gravitational masses of the more-massive (1) and less-massive (2) components, M = m1 + m2 is the total mass, S1,2 are the spin angular momentum vectors of the black holes in the binary, L is the orbital angular momentum vector, assumed to point in the direction, and χ1,2 are the corresponding dimensionless projections of the individual black hole spins. Because the dimensionless spin parameterof each black hole is bounded by 0 ≤ a1,2 < 1, the projections along the orbital axis are bounded by −1 < χ1,2 < 1, and −1<χeff < 1.
We form the population distributions of χeff shown in Fig. 2 by assuming that each black hole in a binary has a dimensionless spin magnitude drawn from one of three distributions, referred to as ‘low’, ‘flat’ and ‘high’:These distributions are shown in Extended Data Fig. 1.
Although 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 because all of the models considered are based on several additional assumptions, such as decoupled distributions for spin magnitude and spin misalignment angle, 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 binary black hole spin–orbit misalignments observed by LIGO is a mixture of binaries with aligned spins and binaries with isotropic spins.
We fit a mixture model32 (labelled model ‘M’ in Fig. 3) in which a fraction fi of binary black holes have spins drawn from an isotropic distribution and a fraction 1 − fi have their spins aligned with the orbital angular momentum. We assume a flat prior on the fraction fi. To test the robustness of our result, we vary the distribution that we assume for the black hole spin magnitude distribution, as with the aligned and isotropic models. We use the flat, high and low distributions (equation (1)), assuming that all black holes have their spin magnitude drawn from the same distribution for both the aligned and isotropic populations. We calculate and plot the posterior on fi given by equation (2) (fi = λ in the derivation) in Extended Data Fig. 2. We find that the mean fraction of binary black holes 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 fi = 1. Therefore, for these spin magnitude distributions we find that the current LIGO O1 and GW170104 observations constrain the majority of binary black holes to have their spins drawn from an isotropic distribution. The odds ratios of these mixture models to the isotropic distribution with low, flat and high spin magnitude models are 0.43, 0.20 and 0.10, respectively. We therefore cannot rule out a mixture with the current data. If several different components contribute substantially to the true spin distribution, then it may take tens to hundreds of detections to accurately determine the mixing fraction, depending on the distribution of spin magnitudes17,32.
LIGO measures χeff better than any other spin parameter, but still with substantial uncertainty33, so we need to properly incorporate measurement uncertainty in our analysis; our analysis must therefore be hierarchical34,35. 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 λ; the system is then observed, represented by the likelihood function, which results in a distribution for the true effective spin (and all other parameters describing the system) that is consistent with the data. The combined joint posterior on the parameters and population parameters λ of each system, implied by a set of observations each with data di, isThe components of this formula are:
• The gravitational wave (marginal) likelihood, p(d | χeff). Here we use ‘marginal’ because we are (implicitly) integrating over all parameters of the signal except χeff. It is the likelihood rather than the posterior that matters for the hierarchical analysis; if we are given posterior distributions or posterior samples, then we need to re-weight to ‘remove’ the prior and obtain the likelihood.
• The population distribution for χeff, p(χeff | λ). This function can be parameterized by population-level parameters λ. (In the cases discussed above, there are no parameters for the population.)
• The prior on the population-level parameters, p(λ).
If we do not care about the χeff parameters for individual events, then we can integrate them out, obtainingIf we are given posterior samples of (i labels the event, j labels the particular posterior sample), drawn from an analysis using a prior p(χeff), then we can approximate the integral by a re-weighted average of the population distribution over the samples (here is the prior used to produce the posterior samples):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. On the basis of Fig. 2, we approximate the χeff distribution of the isotropic population as uniform on χeff ∈ [−0.25, 0.25] and that of the aligned population as uniform on χeff ∈ [0, 0.5]. The odds ratio between the isotropic and aligned models for each event is thenwhere P(A ≤ χeff ≤ B) is the posterior probability (here used to approximate the likelihood) that χeff is between A and B. Our approximations to the χeff posteriors described above produce 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 in the main text. 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 χeff are about 0.1, then the geometric mean of the contribution of each subsequent measurement to the overall odds is approximately 3. After 10 additional events, the odds ratio becomes 5 × 310 ≈ 3 × 105, or 4.6σ, consistent with the results of the more detailed calculation. If the true distribution of spins becomes half as wide (χeff ∈ [−0.125, 0.125] for isotropic and χeff ∈ [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 2.7σ. With small spin magnitudes, our angular resolving power vanishes, as discussed in more detail in Methods section ‘Effect of small spin magnitudes’.
Accumulation of evidence. In Extended Data Table 1 we show how the evidence for an isotropic distribution increases when including: only the two confirmed events (GW150914 and GW151226) from LIGO O1; all LIGO O1 events (including LVT151012); and all four likely binary black hole mergers, including GW170104.
Effect of small spin magnitudes
We consider three models for black hole spin magnitudes in equation (1): low, flat and high. These models were intended to capture some of the uncertainty regarding the black hole spin magnitude distribution. We may remain observationally uncertain about the spin magnitude distribution until we have observations36,37.
Here we extend the low model as:Choosing α = 0 recovers the flat distribution, whereas α = 1 recovers the low distribution. For higher values of α, this distribution becomes more peaked towards a = 0.
In Extended Data Fig. 3 we plot the ratio of isotropic to aligned distributions (plotted as the equivalent σ) with spin magnitudes given by this model with α in the range 0–6. The top axis shows the mean spin magnitude that a given value of α corresponds to (for example, for the flat distribution (α = 0), the mean spin magnitude is 0.5). If typical black hole spins are less than about 0.2, then we have no evidence for one model over the other.
Extended Data Fig. 4 shows the distributions of χeff that would be obtained with a mass ratio q = m2/m1 = 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 the primarily sensitivity is to the changing sign of χeff 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 10 additional detections is quite weak compared to the aligned/isotropic distinction in Fig. 4. The differences in the χeff distribution between q = 1 and q = 0.5 are even smaller than the differences between the different magnitude distributions.
Approximations in the gravitational waveform and selection effects
Although the Advanced LIGO searches use spin-aligned templates, they are efficient in detecting misaligned binary black hole systems38; we assume here that the χeff distribution of observed sources follows the true population.
The model waveforms used to infer the χeff of the three LIGO events incorporate approximations to the true behaviour of the merging systems that are expected to break down for sufficiently high misaligned spins. The effect of these approximations on inference on the parameters describing GW150914 has been investigated in detail39. For this source, statistical uncertainties dominate over any waveform systematics. Detailed comparisons with numerical relativity computations using no approximations to the dynamics40 also suggest that statistical uncertainties dominate the systematics for this system. Systematics may dominate for signals with such a large signal-to-noise ratio (approximately 23) when the source is edge-on or has high spins39. The other two events discussed here have much lower signal-to-noise ratios, 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 sources detected in the future that have high signal-to-noise ratios, are edge-on or have high spin.
Precision of χ eff measurements
Throughout this work we have made the simplifying assumption that the precision to which χeff can be constrained for individual binaries is independent of the properties of the binary. In practice, our ability to constrain χeff is dependent on the properties of the system, in particular its true χeff and mass ratio, which we illustrate in Extended Data Fig. 5.
For this figure, a detected population of 500 binaries was selected from a population with component masses distributed uniformly between 1M⊙ and 30M⊙ (where M⊙ is the mass of the Sun), with m1 + m2 < 30M⊙, with locations distributed uniformly in volume, and with orientations distributed isotropically. We qualify a system as ‘detected’ if it produces a signal-to-noise ratio of more than 8 in the second-loudest detector, to select only coincident events. Data were simulated for each binary, and posteriors were estimated using the LIGO–Virgo parameter estimation library LALInference33 using inspiral-only waveform models (merger and ringdown effects can provide additional information for some binaries, but we ignore those effects here). χeff is better constrained for binaries with high effective spins and high (approximately equal) mass ratios.
We do not expect these effects to qualitatively affect out conclusions, although they could affect predictions for the total number of detections that are necessary to constrain the population. For example, if the Universe preferentially forms asymmetric binaries with low mass ratios, then individual χeff constraints will be systematically worse, requiring more binaries to infer the properties of the population.
No statistical methods were used to predetermine sample size. The choice of 10 additional simulated detections is based on the expected rate of detections over the next few years, not any assumptions about statistical power. Fig. 4 demonstrates that the actual rate of detections gives sufficient statistical power to discriminate between the different models used here.
This analysis used the Julia language41, Python libraries NumPy and SciPy42,43 and the plotting library Matplotlib44, and computations were performed in IPython notebooks45. A repository containing the code and notebooks used for this analysis can be found under an open-source ‘MIT’ license at https://github.com/farr/AlignedVersusIsoSpin.
The datasets generated and analysed during this study are available at https://github.com/farr/AlignedVersusIsoSpin.
We thank R. O’Shaughnessy, C. Berry, D. Gerosa and S. Vitale for discussions and comments on this work. W.M.F., S.S., I.M. and A.V. were supported in part by the STFC. M.C.M. acknowledges support of the University of Birmingham Institute for Advanced Study Distinguished Visiting Fellows programme. S.S. and I.M. acknowledge support from the National Science Foundation under grant number NSF PHY11-25915.