Distribution of effective spins and masses of binary black holes from the LIGO and Virgo O1–O3a observing runs

The distribution of effective spin ${\chi }_{\mathrm{eff}}$, a parameter that encodes the degree of spin-orbit alignment in a binary system, has been widely regarded as a robust discriminator between the isolated and dynamical formation pathways for merging binary black holes. Until the recent release of the GWTC-2 catalog, such tests have yielded inconclusive results due to the small number of events with measurable nonzero spins. In this work, we study the ${\chi }_{\mathrm{eff}}$ distribution of the binary black holes detected in the LIGO-Virgo O1–O3a observing runs. Our focus is on the degree to which the ${\chi }_{\mathrm{eff}}$ distribution is symmetric about ${\chi }_{\mathrm{eff}}=0$ and whether the data provide support for a population of negative-${\chi }_{\mathrm{eff}}$ systems. We find that the ${\chi }_{\mathrm{eff}}$ distribution is asymmetric at 95% credibility, with an excess of aligned-spin binary systems (${\chi }_{\mathrm{eff}}>0$) over antialigned ones. Moreover, we find that there is no evidence for negative-${\chi }_{\mathrm{eff}}$ systems in the current population of binary black holes. Thus, based solely on the ${\chi }_{\mathrm{eff}}$ distribution, dynamical formation is disfavored as being responsible for the entirety of the observed merging binary black holes, while isolated formation remains viable. We also study the mass distribution of the current binary black hole population, confirming that a single truncated power-law distribution in the primary source-frame mass, $m_{1\mathrm{s}}$, fails to describe the observations. Instead, we find that the preferred models have a steep feature at $m_{1\mathrm{s}}\sim 40{\mathrm{M}}_{\odot }$ consistent with a step and an extended, shallow tail to high masses.

Figure 1

Empirical spin distributions of the events in the gold sample (see the Appendix). For each event, parameter estimation samples were obtained using a uniform prior in ${\chi }_{\mathrm{eff}}$. To avoid clutter, event names were abbreviated when this did not cause ambiguity. Left panel: mean effective spin scaled by the standard deviation for each event’s posterior. We see that about $N_0\approx 20$ events in the gold sample are consistent with noisy measurements of a ${\chi }_{\mathrm{eff}}=0$ subpopulation, but a tail in the positive ${\chi }_{\mathrm{eff}}$ end of the distribution is clearly needed in order to accommodate the remaining $\approx 10$ events. Conversely, no such tail seems to be needed at the negative end. Middle panel: ${\chi }_{\mathrm{eff}}$ distribution, where markers and error bars indicate mean and standard deviation. In the cumulative, each event is weighted by the ratio of the event’s sensitive volume to a similar event with zero spins in order to cancel spin selection effects. We see that there are several events with small but well-measured ${\chi }_{\mathrm{eff}}>0$ for which spin selection effects are not important. Right panel: ratio of observed ${\chi }_{\mathrm{eff}}$ to its characteristic value if strong tides were acted on the secondary (blue circles) or primary (orange triangles) black hole progenitor. A number of events are inconsistent with any of these variables being 0 or 1, thereby excluding the strong-tide model as the only mechanism generating black hole spins.

Figure 2

Sketch of the functional form Eq. (6), which we use to parametrize the ${\chi }_{\mathrm{eff}}$ distribution as the sum of three subpopulations. These subpopulations have positive, negative or zero effective spins, with each described by truncated Gaussians that peak at ${\chi }_{\mathrm{eff}}=0$. We use three independent shape parameters: the width of the positive and negative distributions, ${\sigma }_{{\chi }_{\mathrm{eff}}}$, which are constrained to be equal, and the three branching ratios ${\zeta }_j$ which sum to unity. For technical reasons, we fix the width of the ${\chi }_{\mathrm{eff}}\approx 0$ subpopulation to have a small but nonvanishing dispersion of ${\sigma }_0=0.04$.

Figure 3

Constraints on the model parameters of the population model equation (6). We see that a ${\chi }_{\mathrm{eff}}$ distribution symmetric about 0 (black dashed line), with ${\zeta }_{\mathrm{pos}}={\zeta }_{\mathrm{neg}}$, is disfavored by the data. In addition, the population is consistent with having no negative-spin subpopulation. The two-dimensional contours enclose the 50% and 90% credible regions. Parameter values (median and 90% confidence level) are reported for the $\mathrm{GWTC}\text{−}1+\mathrm{IAS}+\text{Gold}$ O3a sample.

Figure 4

Constraints assuming a Gaussian model for the ${\chi }_{\mathrm{eff}}$ distribution. The black dashed line corresponds to ${\sigma }_{{\chi }_{\mathrm{eff}}}=\overline{{\chi }_{\mathrm{eff}}}$; models above the line have sizable support at negative ${\chi }_{\mathrm{eff}}$. Thus, contrary to Fig. 3, under this model one would conclude that negative ${\chi }_{\mathrm{eff}}$ are present in the population.

Figure 5

Adding a broad subpopulation ${\lambda }_0^{\prime }$ with a fraction $\epsilon =0.05$ of the astrophysical rate affects the inferred parameters of the mass distribution. This is a major effect for the truncated power-law model (a), moderate for the broken power-law model (b), and minor for the power $\mathrm{law}+\text{peak model}$ (c). These constraints are derived using the $\mathrm{GWTC}\text{−}1+\mathrm{IAS}+\text{Gold}$ O3a sample of events; we find similar results with $\mathrm{GWTC}\text{−}1+\mathrm{IAS}+\mathrm{GWTC}\text{−}2$. Repeating the analysis with $\epsilon =0.1$ yields similar results as with $\epsilon =0.05$, corroborating that it is the freedom to accommodate outliers that drives these changes rather than the particular choice of $\epsilon$.

Figure 6

Mass distribution predicted by the models favored by the data: truncated power law or broken power law, in both cases with an additional broad subpopulation with a fraction $\epsilon =0.05$ of the total rate responsible for the shallow tail to high masses. These distributions feature a steepening around $40{\mathrm{M}}_{\odot }$, consistent with a step, and a flattening at higher mass. Note that we do not fit for the power-law index or the normalization of the high-mass tail.