Determining the population properties of spinning black holes

There are at least two formation scenarios consistent with the first gravitational-wave observations of binary black hole mergers. In field models, black hole binaries are formed from stellar binaries that may undergo common envelope evolution. In dynamic models, black hole binaries are formed through capture events in globular clusters. Both classes of models are subject to significant theoretical uncertainties. Nonetheless, the conventional wisdom holds that the distribution of spin orientations of dynamically merging black holes is nearly isotropic while field-model black holes prefer to spin in alignment with the orbital angular momentum. We present a framework in which observations of black hole mergers can be used to measure ensemble properties of black hole spin such as the typical black hole spin misalignment. We show how to obtain constraints on population hyperparameters using minimal assumptions so that the results are not strongly dependent on the uncertain physics of formation models. These data-driven constraints will facilitate tests of theoretical models and help determine the formation history of binary black holes using information encoded in their observed spins. We demonstrate that the ensemble properties of binary detections can be used to search for and characterize the properties of two distinct populations of black hole mergers.

Figure 1

The distribution of $z$ for our field model proxy with varying $\sigma$; see Eq. (3). By sending $\sigma \to 0$, we obtain perfect alignment and by sending $\sigma \to \infty$, we obtain an isotropic distribution.

Figure 2

Simulated spin misalignment parameters $(z_1,z_2)$ for the different populations of binary black holes used in our study. Red diamonds are drawn from the isotropic distribution $p_0$ while the blue circles are drawn from the aligned distribution, $p_1(z_1,z_2|{\sigma }_1,{\sigma }_2=0.3,0.5)$; see Eq. (3).

Figure 3

In each panel we plot $1\sigma$ (dark shading), $2\sigma$ (medium shading), and $3\sigma$ (light shading) confidence intervals for different hyperparameters as a function of the number of events $N$. Each column represents a different hyperparameter: ${\sigma }_1$ (left), ${\sigma }_2$ (middle), and $\xi$ (right). Each row represents a different universe; see Table 1. From top to bottom, the universes are A, B, C, D, and E. The dashed line indicates the true hyperparameter values. Note that ${\sigma }_1$ and ${\sigma }_2$ are not defined for universe A, which consists entirely of a population of dynamical mergers. Thus, the first two panels in the top row do not provide meaningful information. They simply show random fluctuations induced by cosmic variance (randomness in the realisations of black hole tilt angles).

Figure 4

Log Bayes factors as a function of the number of GW150914-like events. The dot-dashed red line shows $B_{\mathrm{dyn}}^{\text{field}}$ comparing the pure-field hypothesis to the pure-dynamical hypothesis. The dashed green line shows $B_{\mathrm{dyn}}^{\mathrm{mix}}$ comparing the two-population hypothesis to the dynamical hypothesis. The solid blue line shows $B_{\text{field}}^{\mathrm{mix}}$ comparing the two-population hypothesis to the pure-field hypothesis. The dashed black line denotes $|\ln (B)|=8$, our threshold for distinguishing between models. Each panel is a different universe. The top panel is universe C (equal mixture of field and dynamic). $\text{With}\lesssim 40$ events, there is a strong preference for the two-component hypothesis over the pure-dynamic hypothesis. After $\sim 50$ events there is a preference for the two-component hypothesis over the pure-field hypothesis. The center panel is universe D (majority field with some dynamic). $\text{With}\lesssim 10$ events, there is a strong preference for the two-component and pure-field hypotheses over the pure-dynamic hypothesis. There is a preference for the correct two-population hypothesis over the pure-field hypothesis. The bottom panel is universe E (pure field). $\text{With}\lesssim 10$ events, there is a strong preference for the two-component and field hypotheses over the dynamic hypothesis. There is a marginal preference for the correct field hypothesis over the two-population hypothesis.

Figure 5

Spin maps: maps of posterior spin orientation probability density averaged over many realizations. The latitude is the spin tilt of the primary (more massive) black hole. The longitude is the angle between the projection of the black hole spins onto the orbital plane. The color bar is the number of posterior samples per $5{\mathrm{deg}}^2$ healpix bin. The left panel shows a spin map for 80 events drawn from universe A (see Table 1) in which every binary merges dynamically. The right panel shows 80 events drawn from universe C in which, on average, 50% of the events are drawn from the dynamical population while 50% are drawn from the field population with $({\sigma }_1,{\sigma }_2)=(0.3,0.5)$; see, Eq. (3). The presence of a preferentially oriented population is seen as clustering around the north pole.