We describe a Bayesian formalism for analyzing individual gravitational-wave events in light of the rest of an observed population. This analysis reveals how the idea of a ``population-informed prior arises naturally from a suitable marginalization o
f an underlying hierarchical Bayesian model which consistently accounts for selection effects. Our formalism naturally leads to the presence of ``leave-one-out distributions which include subsets of events. This differs from other approximations, also known as empirical Bayes methods, which effectively double count one or more events. We design a double-reweighting post-processing strategy that uses only existing data products to reconstruct the resulting population-informed posterior distributions. Although the correction we highlight is an important conceptual point, we find it has a limited impact on the current catalog of gravitational-wave events. Our approach further allows us to study, for the first time in the gravitational-wave literature, correlations between the parameters of individual events and those of the population.
Following the collapse of their cores, some of the massive binary stars that populate our Universe are expected to form merging binaries composed of black holes and neutron stars. Gravitational-wave observations of the resulting compact binaries can
reveal precious details on the inner workings of the supernova mechanism and the subsequent formation of compact objects. Within the framework of the population-synthesis code MOBSE, we present the implementation of a new supernova model that relies on the compactness of the collapsing star. The model has two free parameters, namely the compactness threshold that separates the formation of black holes and that of neutron stars, and the fraction of the envelope that falls back onto the newly formed black holes. We compare this model extensively against other prescriptions that are commonly used in binary population synthesis. We find that the cleanest signatures of the role of the pre-supernova stellar compactness are (i) the relative formation rates of the different kinds of compact binaries, which mainly depend on the compactness threshold parameter, and (ii) the location of the upper edge of the mass gap between the lightest black holes and the heaviest neutron stars, which mainly depends on the fallback fraction.
Solutions to the two-body problem in general relativity allow us to predict the mass, spin and recoil velocity of a black-hole merger remnant given the masses and spins of its binary progenitors. In this paper we address the inverse problem: given a
binary black-hole merger, can we use the parameters measured by gravitational-wave interferometers to tell if the binary components are of hierarchical origin, i.e. if they are themselves remnants of previous mergers? If so, can we determine at least some of the properties of their parents? This inverse problem is in general overdetermined. We show that hierarchical mergers occupy a characteristic region in the plane composed of the effective spin parameters $chi_{rm eff}$ and $chi_{rm p}$, and therefore a measurement of these parameters can add weight to the hierarchical-merger interpretation of some gravitational-wave events, including GW190521. If one of the binary components has hierarchical origin and its spin magnitude is well measured, we derive exclusion regions on the properties of its parents: for example we infer that the parents of GW190412 (if hierarchical) must have had unequal masses and low spins. Our formalism is quite general, and it can be used to infer constraints on the astrophysical environment producing hierarchical mergers.
We review theoretical findings, astrophysical modeling, and current gravitational-wave evidence of hierarchical stellar-mass black-hole mergers. While most of the compact binary mergers detected by LIGO and Virgo are expected to consist of first-gene
ration black holes formed from the collapse of stars, others might instead be of second (or higher) generation, containing the remnants of previous black-hole mergers. Such a subpopulation of hierarchically assembled black holes presents distinctive gravitational-wave signatures, namely higher masses, possibly within the pair-instability mass gap, and dimensionless spins clustered at the characteristic value of $sim$0.7. In order to produce hierarchical mergers, astrophysical environments need to overcome the relativistic recoils imparted to black-hole merger remnants, a condition which prefers hosts with escape speeds $gtrsim$ 100 km/s. Promising locations for efficient production of hierarchical mergers include nuclear star clusters and accretion disks surrounding active galactic nuclei, though environments that are less efficient at retaining merger products such as globular clusters may still contribute significantly to the detectable population of repeated mergers. While GW190521 is the single most promising hierarchical-merger candidate to date, constraints coming from large population analyses are becoming increasingly more powerful.
The occurrence of pair-instability supernovae is predicted to prevent the formation of black holes with masses $gtrsim 50 M_odot$. Recent gravitational-wave detections in this mass range require an explanation beyond that of standard stellar collapse
. Current modeling strategies include the hierarchical assembly of previous generations of black-hole mergers as well as other mechanisms of astrophysical nature (lowered nuclear-reaction rates, envelope retention, stellar mergers, accretion, dredge-up episodes). In this paper, we point out the occurrence of an exclusion region that cannot be easily populated by hierarchical black-hole mergers. A future gravitational-wave detection of a black hole with mass $gtrsim 50M_odot$ and spin $lesssim 0.2$ will indicate that the pair-instability mass gap is polluted in some other way. Such a putative outlier can be explained using hierarchical mergers only with considerable fine-tuning of both mass ratio and spins of the preceding black-hole merger -- an assumption that can then be cross-checked against the bulk of the gravitational-wave catalog.
Binary black holes with spins that are aligned with the orbital angular momentum do not precess. However, post-Newtonian calculations predict that up-down binaries, in which the spin of the heavier (lighter) black hole is aligned (antialigned) with t
he orbital angular momentum, are unstable when the spins are slightly perturbed from perfect alignment. This instability provides a possible mechanism for the formation of precessing binaries in environments where sources are preferentially formed with (anti) aligned spins. In this paper, we present the first full numerical relativity simulations capturing this instability. These simulations span $sim 100$ orbits and $sim 3$-$5$ precession cycles before merger, making them some of the longest numerical relativity simulations to date. Initialized with a small perturbation of $1^{circ}$-$10^{circ}$, the instability causes a dramatic growth of the spin misalignments, which can reach $sim 90^{circ}$ near merger. We show that this leaves a strong imprint on the subdominant modes of the gravitational wave signal, which can potentially be used to distinguish up-down binaries from other sources. Finally, we show that post-Newtonian and effective-one-body approximants are able to reproduce the unstable dynamics of up-down binaries extracted from numerical relativity.
Shortly after a new class of objects is discovered, the attention shifts from the properties of the individual sources to the question of their origin: do all sources come from the same underlying population, or several populations are required? What
are the properties of these populations? As the detection of gravitational waves is becoming routine and the size of the event catalog increases, finer and finer details of the astrophysical distribution of compact binaries are now within our grasp. This Chapter presents a pedagogical introduction to the main statistical tool required for these analyses: hierarchical Bayesian inference in the presence of selection effects. All key equations are obtained from first principles, followed by two examples of increasing complexity. Although many remarks made in this Chapter refer to gravitational-wave astronomy, the write-up is generic enough to be useful to researchers and graduate students from other fields.
Two of the dominant channels to produce the black-hole binary mergers observed by LIGO and Virgo are believed to be the isolated evolution of stellar binaries in the field and dynamical formation in star clusters. Their relative efficiency can be cha
racterized by a mixing fraction. Pair instabilities prevent stellar collapse from generating black holes more massive than about $45 M_odot$. This mass gap only applies to the field formation scenario, and it can be filled by repeated mergers in clusters. A similar reasoning applies to the binarys effective spin. If black holes are born slowly rotating, the high-spin portion of the parameter space (the spin gap) can only be populated by black hole binaries that were assembled dynamically. Using a semianalytical cluster model, we show that future gravitational-wave events in either the mass gap, the spin gap, or both can be leveraged to infer the mixing fraction between the field and cluster formation channels.
We report on advances to interpret current and future gravitational-wave events in light of astrophysical simulations. A machine-learning emulator is trained on numerical population-synthesis predictions and inserted into a Bayesian hierarchical fram
ework. In this case study, a modest but state-of-the-art suite of simulations of isolated binary stars is interpolated across two event parameters and one population parameter. The validation process of our pipelines highlights how omitting some of the event parameters might cause errors in estimating selection effects, which propagates as systematics to the final population inference. Using LIGO/Virgo data from O1 and O2 we infer that black holes in binaries are most likely to receive natal kicks with one-dimensional velocity dispersion $sigma$ = 105+44 km/s. Our results showcase potential applications of machine-learning tools in conjunction with population-synthesis simulations and gravitational-wave data.
Binary black hole mergers are of great interest to the astrophysics community, not least because of their promise to test general relativity in the highly dynamic, strong field regime. Detections of gravitational waves from these sources by LIGO and
Virgo have garnered widespread media and public attention. Among these sources, precessing systems (with misaligned black-hole spin/orbital angular momentum) are of particular interest because of the rich dynamics they offer. However, these systems are, in turn, more complex compared to nonprecessing systems, making them harder to model or develop intuition about. Visualizations of numerical simulations of precessing systems provide a means to understand and gain insights about these systems. However, since these simulations are very expensive, they can only be performed at a small number of points in parameter space. We present binaryBHexp, a tool that makes use of surrogate models of numerical simulations to generate on-the-fly interactive visualizations of precessing binary black holes. These visualizations can be generated in a few seconds, and at any point in the 7-dimensional parameter space of the underlying surrogate models. With illustrative examples, we demonstrate how this tool can be used to learn about precessing binary black hole systems.
