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Title: Demographics of Hierarchical Black Hole Mergers in Dense Star Clusters
Authors: Giacomo Fragione and Frederic A. Rasio
First Author’s Institution: Northwestern University
Status: Published in ApJ
Observations of gravitational waves from binary black hole mergers made by LIGO, Virgo, and KAGRA have revolutionized our understanding of the demographics of compact objects like neutron stars and black holes. At the range detectable by these instruments, for example, they have shed light on the distribution of black hole masses, which gives us a glimpse into the late-stage evolution of massive stars. In particular, stellar evolution models predict a dearth of black hole remnants with masses between fifty and a few hundred times the mass of our Sun, because of the so-called pair-instability process. However, LIGO–Virgo–KAGRA observations have indicated the existence of black hole binaries where one member of the binary is in this pair-instability range, meaning that the simple stellar evolution picture doesn’t fully explain the observed population!
Binary black holes are believed to form by two main channels: in isolation, from a binary star system, and dynamically, as depicted in Figure 1. In the former (left), the binary system evolves through a common-envelope phase, which, if the conditions are right, will result in a binary black hole system. In the dynamical pathway (right), interactions between stars in the dense centers of clusters are frequent and can lead to the formation of binary black hole systems. Subsequent interactions can tighten the orbit, leading to mergers.
Figure 1: Cartoon depiction of the two primary channels for binary black hole formation (left and right columns), with time increasing from top to bottom in each column. At left, a binary star system enters a common-envelope phase that shrinks the orbit of the binary. If this results in ejection of the envelope, the system will eventually form a binary black hole system that will ultimately merge. At right, a binary system of a black hole and a star will become a system of binary black holes through a three-body interaction in a dense star cluster. Later interactions will shrink the orbit until the two black holes merge. [Mapelli 2020; CC BY 4.0]
Demographics of Cluster Hosts
The primary challenge associated with this channel of intermediate-mass black hole formation is the “recoil kick” that results from asymmetry in the merger process. In some cases, the velocity with which the resulting black hole forms can exceed the escape velocity of the star cluster in which it formed, leading to ejection (and thus limiting the possibility of a subsequent merger). Using the modeling framework developed in a previous work, today’s authors are able to predict the properties of clusters that will be able to retain a binary black hole remnant after the merger, as displayed in Figure 2. From these panels it is clear that hierarchical black holes are likely only produced and retained in the most massive and densest clusters, as these will be the ones with the deepest potential wells and highest rates of interaction.
Figure 2: The probability that a merger remnant will survive in the cluster (color scale) as a function of cluster mass and density, going from the merger of two first-generation black holes (left), to the merger of a first- and second-generation black hole (middle), and the merger of two second-generation black holes (right). The points correspond to observations of different types of star clusters from the literature. Notice how the remnant of two first-generation black holes merging is relatively easy to retain, but it becomes progressively more difficult as we introduce later generations. [Adapted from Fragione and Rasio 2023]
Merger Rates and Assorted Sundries
With their framework in hand and the properties of the cluster hosts understood, today’s authors then average over the distribution of star clusters of different masses as a function of time to predict merger rates of various generations of hierarchical black holes. In Figure 3, the authors demonstrate that massive clusters (with masses up to 107 solar masses) are necessary to produce later generations of black hole mergers. That is, they find that the rates of hierarchical black hole mergers fall as the maximum cluster mass is lowered from 10 million to 1 million solar masses, as the lower-mass clusters are less likely to hold onto the resulting black holes. In kind, the merger rates for component black holes later than third-generation disappear.
Figure 3: The merger rates as a function of cosmological redshift (i.e., over time in the universe’s history) for combinations of merging black holes of different generations (e.g., 1G refers to a first-generation black hole). The left panel is the prediction including star clusters with masses up to 10 million solar masses, while the right panel only includes clusters up to 1 million solar masses. There are no mergers beyond 3G in the right panel, demonstrating that only the most massive clusters can host late-generation black-hole mergers. [Adapted from Fragione and Rasio 2023]
These results, while preliminary, can be extended (by incorporating other metrics, such as the black hole spins) to statistically measure the likelihood of individual gravitational wave events being associated with hierarchical mergers. However, they represent an exciting step towards understanding where these intermediate-mass black holes come from and provide a compelling, natural explanation for how a large population of massive black holes can form and continue to evolve over cosmic time!
Original astrobite edited by Mark Popinchalk.
About the author, Sahil Hegde:
I am an astrophysics PhD student at UCLA working on using semi-analytic models to study the formation of the first stars and galaxies in the universe. I completed my undergraduate at Columbia University, and am originally from the San Francisco Bay Area. Outside of astronomy you’ll find me playing tennis, surfing (read: wiping out), and playing board games/TTRPGs!