Using spin to understand the formation of LIGO's black holes

Ben Farr, Daniel E. Holz, Will M. Farr

I Introduction

The direct detection of gravitational waves (GWs) from binary black holes (BBHs) has become almost routine, with three confident events (GW150914 (Abbott et al. 2016a), GW151226 (Abbott et al. 2016b), and GW170104 (Abbott et al. 2017)) and one candidate event LVT151012 (Abbott et al. 2016c) identified by the Advanced LIGO detectors. The GW signatures of these binaries encode properties of the binary (Veitch et al. 2015), and in particular can be used to measure the spin properties of the binary’s black holes (BHs). The astrophysical processes by which these systems form remain uncertain; two generic formation channels include the evolution of an isolated pair of stars that were born together (i.e., “in the field”) (Tutukov and Yungelson 1993; Belczynski et al. 2016; Stevenson et al. 2017a; Mandel and de Mink 2016a; Marchant et al. 2016a), and dynamical interactions in a dense stellar environment (i.e., globular clusters) (Sigurdsson and Hernquist 1993a; Portegies Zwart and McMillan 2000; Rodriguez et al. 2015a; Rodriguez et al. 2016).

Computational models of these formation channels provide predictions for the rates and distributions of binary masses. However, these predictions are highly dependent upon assumptions about poorly understood processes (e.g., common envelope evolution). More robust are the predictions for binary spin properties, particularly the orientation of BH spins with respect to the orbital angular momentum of the binary. A generic characteristic of dynamical formation is that the spin orientations of the component black holes are isotropic with respect to the orbital angular momentum Sigurdsson and Hernquist 1993b; Portegies Zwart and McMillan 2000; Rodriguez et al. 2015b; Stone et al. 2017; Rodriguez et al. 2016; Schnittman 2004; Bogdanović et al. 2007. Isolated binaries, on the other hand, are generally expected to be preferentially aligned with the orbital angular momentum Tutukov and Yungelson 1993; Belczynski et al. 2016; Stevenson et al. 2017b; Mandel and de Mink 2016b; Marchant et al. 2016b.This conclusion can be weakened through the impact of natal kicks O’Shaughnessy et al. 2017, but it is difficult for the isolated channel to produce a significant fraction binary mergers with large misalignment ( $\hat{S}\cdot\hat{L}<0$ , where $\hat{S}$ and $\hat{L}$ point along the stellar spin and orbital angular momentum vectors, respectively). In the absence of firm predictions for the full spin distributions (orientation and magnitude) from various formation channels, we provide a model-agnostic approach specially suited for looking for spin isotropy in the BBH population.

where $M=m_{1}+m_{2}$ is the total mass of the system, and $-1<\chi_{1,2}<1$ are the projections of the spin vectors of each component BH along the orbital axis. As pointed out in (Farr et al. 2017), by measuring the effective spin distribution of the population we can infer misalignment characteristics, and consequently formation channels.

II Methods

II.2 Hierarchical Population Inference

Ultimately we would like to calculate $p(\lambda\rvert\{d^{i}\})$ , the posterior density function for the parameters $\lambda$ describing our population given the data measured around each event $\{d^{i}\}$ . Since the effective spin cannot be determined precisely form any given observation, we want to marginalize over it, leading to the following expression for the posterior density function for population parameters $\lambda$ :

We will make use of this hierarchical posterior density function throughout this work using various population models.

Figure 2 shows the posterior constraints on $\rho$ for the very-very-low-spin aligned and isotropic populations, where for the aligned population the simulated value ( $\rho=1$ ) is incorrectly excluded with high confidence. This shortcoming, due to the mismatch between the population model and the simulated population, is addressed in the following subsection.

II.5 Constraining Component Spins

III Results from LIGO’s BBHs

If we now assume a particular formation scenario for LIGO’s BBHs, we can infer the BH spin magnitude distribution following the approach in Section II.5. Figure 7 shows the posterior constraints on the spin magnitude distribution assuming all of LIGO’s BBHs were dynamically formed. Figure 8 shows the posterior constraints on the spin magnitude distribution assuming all of LIGO’s BBHs are aligned, the currently preferred scenario. If the population is indeed aligned, then even with just the 4 candidate BBH mergers detected to date we can already say that there are likely more low spin systems ( $a\lesssim 1/3$ ) than moderate ( $1/3\lesssim a\lesssim 2/3$ ) or high ( $a\gtrsim 2/3$ ) spin systems.

IV Conclusions

With the 4 likely BBH systems observed by LIGO thus far we find that an aligned formation scenario (i.e., isolated or field formation) is slightly preferred over an isotropic scenario (i.e., dynamical), with an odds ratio in favor of alignment of 1.1. Similarly to Farr et al. 2017, but with a more general model, we find that if all of LIGO’s BBHs are assumed to come from this aligned population, then most BH spins must be low ( $a\lesssim 1/3$ ).