Recent work paints a conflicting portrait of the distribution of black hole spins in merging binaries measured with gravitational waves. Some analyses find that a significant fraction of merging binaries contain at least one black hole with a spin ti
lt $>90^circ$ with respect to the orbital angular momentum vector, which has been interpreted as a signature for dynamical assembly. Other analyses find the data are consistent with a bimodal population in which some binaries contain black holes with negligible spin while the rest contain black holes with spin vectors preferentially aligned with the orbital angular momentum vector. In this work, we scrutinize models for the distribution of black hole spins to pinpoint possible failure modes in which the model yields a faulty conclusion. We reanalyze data from the second LIGO--Virgo gravitational-wave transient catalog (GWTC-2) using a revised spin model, which allows for a sub-population of black holes with negligible spins. In agreement with recent results by Roulet et al., we show that the GWTC-2 detections are consistent with two distinct sub-populations. We estimate that $70-90%$ (90% credible interval) of merging binaries contain black holes with negligible spin $chi approx 0$. The remaining binaries are part of a second sub-population in which the spin vectors are preferentially (but not exactly) aligned to the orbital angular momentum. The black holes in this second sub-population are characterized by spins of $chisim0.5$. We suggest that the inferred spin distribution is consistent with the hypothesis that all merging binaries form via the field formation scenario.
Time series analysis is ubiquitous in many fields of science including gravitational-wave astronomy, where strain time series are analyzed to infer the nature of gravitational-wave sources, e.g., black holes and neutron stars. It is common in gravita
tional-wave transient studies to apply a tapered window function to reduce the effects of spectral artifacts from the sharp edges of data segments. We show that the conventional analysis of tapered data fails to take into account covariance between frequency bins, which arises for all finite time series -- no matter the choice of window function. We discuss the origin of this covariance and show that as the number of gravitational-wave detections grows, and as we gain access to more high signal-to-noise ratio events, this covariance will become a non-negligible source of systematic error. We derive a framework that models the correlation induced by the window function and demonstrate this solution using both data from the first LIGO--Virgo transient catalog and simulated Gaussian noise.
In General Relativity, the spacetimes of black holes have three fundamental properties: (i) they are the same, to lowest order in spin, as the metrics of stellar objects; (ii) they are independent of mass, when expressed in geometric units; and (iii)
they are described by the Kerr metric. In this paper, we quantify the upper bounds on potential black-hole metric deviations imposed by observations of black-hole shadows and of binary black-hole inspirals in order to explore the current experimental limits on possible violations of the last two predictions. We find that both types of experiments provide correlated constraints on deviation parameters that are primarily in the tt-components of the spacetimes, when expressed in areal coordinates. We conclude that, currently, there is no evidence for a deviations from the Kerr metric across the 8 orders of magnitudes in masses and 16 orders in curvatures spanned by the two types of black holes. Moreover, because of the particular masses of black holes in the current sample of gravitational-wave sources, the correlations imposed by the two experiments are aligned and of similar magnitudes when expressed in terms of the far field, post-Newtonian predictions of the metrics. If a future coalescing black-hole binary with two low-mass (e.g., ~3 Msun) components is discovered, the degeneracy between the deviation parameters can be broken by combining the inspiral constraints with those from the black-hole shadow measurements.
We study the population properties of merging binary black holes in the second LIGO--Virgo Gravitational-Wave Transient Catalog assuming they were all formed dynamically in gravitationally bound clusters. Using a phenomenological population model, we
infer the mass and spin distribution of first-generation black holes, while self-consistently accounting for hierarchical mergers. Considering a range of cluster masses, we see compelling evidence for hierarchical mergers in clusters with escape velocities $gtrsim 100~mathrm{km,s^{-1}}$. For our most probable cluster mass, we find that the catalog contains at least one second-generation merger with $99%$ credibility. We find that the hierarchical model is preferred over an alternative model with no hierarchical mergers (Bayes factor $mathcal{B} > 1400$) and that GW190521 is favored to contain two second-generation black holes with odds $mathcal{O}>700$, and GW190519, GW190602, GW190620, and GW190706 are mixed-generation binaries with $mathcal{O} > 10$. However, our results depend strongly on the cluster escape velocity, with more modest evidence for hierarchical mergers when the escape velocity is $lesssim 100~mathrm{km,s^{-1}}$. Assuming that all binary black holes are formed dynamically in globular clusters with escape velocities on the order of tens of $mathrm{km,s^{-1}}$, GW190519 and GW190521 are favored to include a second-generation black hole with odds $mathcal{O}>1$. In this case, we find that $99%$ of black holes from the inferred total population have masses that are less than $49,M_{odot}$, and that this constraint is robust to our choice of prior on the maximum black hole mass.
In dense stellar environments, the merger products of binary black hole mergers may undergo additional mergers. These hierarchical mergers are predicted to have higher masses than the first generation of black holes made from stars. The components of
hierarchical mergers are expected to have significant characteristic spins $chisim 0.7$. However, since the population properties of first-generation black holes are uncertain, it is difficult to know if any given merger is first-generation or hierarchical. We use observations of gravitational waves to reconstruct the binary black hole mass and spin spectrum of a population containing hierarchical merger events. We employ a phenomenological model that captures the properties of merging binary black holes from simulations of dense stellar environments. Inspired by recent work on the isolated formation of low-spin black holes, we include a zero-spin subpopulation. We analyze binary black holes from LIGO and Virgos first two observing runs, and find that this catalog is consistent with having no hierarchical mergers. We find that the most massive system in this catalog, GW170729, is mostly likely a first-generation merger, having a $4%$ probability of being a hierarchical merger assuming a $5 times 10^5 M_{odot}$ globular cluster mass. Using our model, we find that $99%$ of first-generation black holes in coalescing binaries have masses below 44 $M_{odot}$, and the fraction of binaries with near-zero spin is $0.051^{+0.156}_{-0.048}$ ($90%$ credible interval). Upcoming observations will determine if hierarchical mergers are a common source of gravitational waves.
The gravitational waveform of a merging stellar-mass binary is described at leading order by a quadrupolar mode. However, the complete waveform includes higher-order modes, which encode valuable information not accessible from the leading-order mode
alone. Despite this, the majority of astrophysical inferences so far obtained with observations of gravitational waves employ only the leading order mode because calculations with higher-order modes are often computationally challenging. We show how to efficiently incorporate higher-order modes into astrophysical inference calculations with a two step procedure. First, we carry out Bayesian parameter estimation using a computationally cheap leading-order-mode waveform, which provides an initial estimate of binary parameters. Second, we weight the initial estimate using higher-order mode waveforms in order to fold in the extra information from the full waveform. We use mock data to demonstrate the effectiveness of this method. We apply the method to each binary black hole event in the first gravitational-wave transient catalog GWTC-1 to obtain posterior distributions and Bayesian evidence with higher-order modes. Performing Bayesian model selection on the events in GWTC-1, we find only a weak preference for waveforms with higher order modes. We discuss how this method can be generalized to a variety of other applications.
This is an introduction to Bayesian inference with a focus on hierarchical models and hyper-parameters. We write primarily for an audience of Bayesian novices, but we hope to provide useful insights for seasoned veterans as well. Examples are drawn f
rom gravitational-wave astronomy, though we endeavor for the presentation to be understandable to a broader audience. We begin with a review of the fundamentals: likelihoods, priors, and posteriors. Next, we discuss Bayesian evidence, Bayes factors, odds ratios, and model selection. From there, we describe how posteriors are estimated using samplers such as Markov Chain Monte Carlo algorithms and nested sampling. Finally, we generalize the formalism to discuss hyper-parameters and hierarchical models. We include extensive appendices discussing the creation of credible intervals, Gaussian noise, explicit marginalization, posterior predictive distributions, and selection effects.
