Dynamical Evolution of Black Holes in Globular Clusters

The ground-breaking first detection of gravitational waves (GWs) in 2015 by LIGO (Abbott et al. 2016a,b,c,d,e) has sparked extensive theoretical study of the formation of GW sources, in particular binary star systems with black hole (BH) components. Two primary formation channels for these GW-emitting BH binaries are thought to exist: (1) formation from isolated primordial binaries (so-called “field formation”) and (2) dynamical formation in dense environments, such as globular clusters (GCs), which is the focus of my work.

Figure 1: Milky Way globular cluster 47 Tuc (NASA/ESA)

GCs are stellar systems containing hundreds of thousands to millions of stars found in the halos of their host galaxies. The Milky Way (MW) contains approximately 150 observed GCs. Unlike the galactic field, where individual stars and binaries essentially evolve as isolated systems due to the low densities of these environments, GCs are dense environments where gravitational interactions between stars are common. For BHs, which rapidly congregate in their host clusters’ dense cores upon formation due to mass-segregation, the effect of dynamical encounters is even more pronounced. A single BH can experiences tens or even hundreds of gravitational encounters over its lifetime within a GC. Numerical simulations of GCs have shown that the presence of BHs in a GC can significantly influence the way the host cluster evolves, which in turn affects the overall BH dynamics and the formation of binary BH systems that may be observable as gravitational-wave sources.

The study of BHs in GCs is further motivated from an observational perspective. In recent years, an increasing number of stellar-mass BH candidates have been observed within GCs as members of binary systems with luminous companions. The first such systems were observed in mass-transferring configurations and identified through radio and/or X-ray observations (e.g., Strader et al. 2012). To date, mass-transferring binaries with BH-candidate accretors have been identified in four MW GCs and multiple candidates have also been identified in extragalactic GCs (e.g., Maccarone et al. 2007). Most recently, a stellar-mass BH candidate was identified in the MW GC NGC 3201 through radial-velocity observations in a detached binary with a main-sequence companion (Giesers et al. 2018). This observation marks the first detection of a stellar-mass BH candidate through radial-velocity methods.

Understanding in more detail the dynamical evolution of BHs in GCs will be central to the astrophysical interpretation of the hundreds to thousands of merging BBH signals now expected to be detected by LIGO over the next few years (Abbott et al. 2016b,d). Additionally, a better theoretical understanding of the formation channels and expected source properties will help in the development of improved data analysis and detection strategies, and in the design of future GW observatories, including space-based missions such as LISA, which is now becoming a higher priority given the recent successes of LIGO and the LISA Pathfinder mission (e.g., Armano et al. 2016).

To study the dynamical evolution of BHs in GCs, I use Northwestern’s Cluster Monte Carlo code (CMC), which uses Monte Carlo methods to model the evolution of GCs from formation up to present day. CMC incorporates all physics relevant to the evolution of BHs and BH binaries in GCs, including two-body relaxation, single and binary star evolution, three-body binary formation, galactic tides, and small-N gravitational encounters. In collaboration with my various colleagues in CIERA, I’ve pursued several projects relating to the formation and evolution of compact binaries in GCs, summarized below.

Dynamical Formation of Accreting Black Hole Binaries

Figure 2: Formation history of two characteristic BH binaries

In Kremer et al. (2018a), we explored the formation and evolution of accreting BH binaries in GCs. We demonstrated that these mass-transferring systems form through two distinct channels, illustrated in Figure 2: (1) binary-mediated formation, where a combination of dynamical encounters and tidal forces gradually harden the binary to the point of Roche-lobe overflow and (2) triple-mediated formation, where a hierarchical triple is formed the Lidov-Kozai oscillations drive the inner binary to the mass transfer.

Traditionally, it has been argued that finding even a few BH-binary candidates in GCs indicates a much larger population of undetectable BHs retained in those clusters (e.g., Strader et al. 2012). We showed, however, that while BH-non-BH binaries can readily form in GCs in both detached and mass-transferring configurations, the number of BH-non BH binaries retained within a cluster at late times is independent of the total number of BHs retained (see Figure 3). We demonstrated that the formation of BH-non-BH binaries in a GC is a self-regulated process limited by a complex competition between the number of BHs

Figure 3: Number of BH-XRBs versus total number of BHs at late times for set of 40 GC models.

and the number density of non-BH-non-BH binaries in the region of the GC where BHs mix with non-BHs (see Figure 4). This competition ensures that the number of BH-non-BH binaries found in a GC at late times remains independent of the total number of BHs retained in the cluster.

Figure 4: Cartoon illustration of the interplay of total number of BHs and density of “mixing zone” of BHs and non-BHs.





As a follow-up to Kremer et al. (2018a) which studied accreting BH binaries retained in GCs, we explored the accreting BH binaries and accreting NS binaries ejected from their host cluster into the Galactic halo in Kremer et al. (2018b). Such systems may be observed as low-mass X-ray binaries (LMXBs). Using a set of 137 GC models that, overall, effectively match the observed properties of the MW GCs, we explored the contribution of GCs to the population of LMXBs in the Galactic halo. We demonstrated that although several hundred mass-transferring BH and NS binaries have likely been ejected into the halo, only a few are likely to be observed as X-ray sources, on the basis of their duty cycles and X-ray luminosities at present.

How Black Holes Shape Globular Clusters: Modeling NGC 3201

Figure 5: Surface brightness profiles for all GC models at late times compared to NGC 3201 (yellow circles; data from Trager et al. 1995). Black lines show models retaining >200 BHs at late times and blue lines show models retaining < 10 BHs at late times.

In response to the first stellar-mass BH discovered through radial-velocity measurements in the MW GC NGC 3201, Kremer et al. (2018c) used CMC to develop a set of GC models that accurately match the observational features of NGC 3201. We showed that by varying the magnitude of BH natal kicks, the number of retained BHs at late times changes substantially, which in turn, determines the observational features of our GC models at late times (see Figure 5). In particular, we showed that in order to produce a GC model that matches the observational properties (including surface brightness profile, velocity dispersion profile, core radius, and half-light radius) of NGC 3201 at present, the model must retain ~200 BHs or more.

Figure 6: Orbital parameters of all BH-MS binaries found in BH-rich models at late times. The black “x” marks the BH-MS binary observed in NGC 3201.

Additionally, we demonstrated that GC models retaining large numbers of BHs at late times can harbor detached BH–MS binaries similar in properties to the system recently detected through radial-velocity measurements in NGC 3201. Figure 6 shows the orbital parameters of all detached BH-MS found in our models at late times that retain large numbers of BHs at late times and have structural parameters similar to those of NGC 3201. Filled circles mark the 4 BH-MS binaries found in our single “best-fit” model. Open circles show BH-MS binaries found in all other models that retain many BHs. The horizontal dashed line marks the turn-off mass for clusters of this particular age and metallicity. The black “x” marks the quoted values of the MS mass and minimum for the BH-MS binary detected in NGC 3201 (Giesers et al. 2018).


LISA Sources in Milky-Way Globular Clusters

In the coming decades, the space-based GW observatory LISA is expected to revolutionize GW astronomy and provide a wealth of new information pertaining to BHs and other compact objects. In Kremer et al. (2018d), we demonstrated that compact binaries within the LISA sensitivity range are naturally produced in GCs. We showed that the MW GC system contains ~40 compact binaries that will be observable by LISA, including ~10 BH-BH binaries. Furthermore, we showed that several of these BH-BH binaries can have signal-to-noise ratios large enough to be detectable at the distance of the Andromeda galaxy or even the Virgo cluster. 

Figure 7: All compact object binaries found in set of GC models at late times. The background gray region marks the LISA sensitivity range. The three solid black curves mark the boundary for detecting systems with S/N > 2 at distances of d=9 kpc (bottom; median heliocentric distance of MW GCs), d=800 kpc (middle; distance to Andromeda), and d=16 Mpc (top; distance to the Virgo cluster).

Accreting Double White Dwarf Binaries

In addition to the dynamical formation of LISA sources in GCs, I have also studied the formation of LISA sources as isolated binaries in the Galactic field. Double WD (DWD) binaries will make up the largest fraction of close compact-object binaries in the MW (e.g., Marsh et al. 1995). In Kremer et al. (2015), we developed a computational method to calculate the long-term evolution of DWDs accreting through direct impact and showed that a large fraction of these systems undergo stable mass transfer. As a follow-up to this analysis, we explored the application of these systems to LISA in Kremer et al. (2017) and showed that these accreting DWDs may uniquely exhibit prominent chirps.


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