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The discovery of gravitational waves, first observed in September 2015 following the merger of a binary black hole system, has already revolutionised our understanding of the Universe. This was further enhanced in August 2017, when the coalescence of a binary neutron star system was observed both with gravitational waves and a variety of electromagnetic counterparts; this joint observation marked the beginning of gravitational multimessenger astronomy. The Einstein Telescope, a proposed next-generation ground-based gravitational-wave observatory, will dramatically increase the sensitivity to sources: the number of observations of gravitational waves is expected to increase from roughly 100 per year to roughly 100'000 per year, and signals may be visible for hours at a time, given the low frequency cutoff of the planned instrument. This increase in the number of observed events, and the duration with which they are observed, is hugely beneficial to the scientific goals of the community but poses a number of significant computing challenges. Moreover, the currently used computing algorithms do not scale to this new environment, both in terms of the amount of resources required and the speed with which each signal must be characterised. This contribution will discuss the Einstein Telescope's computing challenges, and the activities that are underway to prepare for them. Available computing resources and technologies will greatly evolve in the years ahead, and those working to develop the Einstein Telescope data analysis algorithms will need to take this into account. It will also be important to factor into the initial development of the experiment's computing model the availability of huge parallel HPC systems and ubiquitous Cloud computing; the design of the model will also, for the first time, include the environmental impact as one of the optimisation metrics., Comment: Submitted for the proceedings of CHEP2023

Pre-merger localization of binary neutron stars (BNSs) is one of the most important scientific goals for the third generation (3G) gravitational wave (GW) detectors. It will enable the electromagnetic observation of the whole process of BNS coalescence, especially for the pre-merger and merger phases which have not been observed yet, opening a window for deeper understandings of compact objects. To reach this goal, we describe a novel combination of multi-band matched filtering and semi-analytical localization algorithms to achieve early-warning localization of long BNS signals in 3G detectors. Using our method we are able to efficiently simulate one month of observations with a three-detector 3G network, and show that it is possible to provide accurate sky localizations more than 30 minutes before the merger. Our simulation shows that there could be ~ 10 (~ 100) BNS events localized within 100 deg2, 20 (6) minutes before merger, per month of observation., Comment: 8 pages, 5 figures. Accepted version. Data release available

Along their path from source to observer, gravitational waves may be gravitationally lensed by massive objects. This results in distortions of the observed signal which can be used to extract new information about fundamental physics, astrophysics, and cosmology. Searches for these distortions amongst the observed signals from the current detector network have already been carried out, though there have as yet been no confident detections. However, predictions of the observation rate of lensing suggest detection in the future is a realistic possibility. Therefore, preparations need to be made to thoroughly investigate the candidate lensed signals. In this work, we present some of the follow-up analyses and strategies that could be applied to assess the significance of such events and ascertain what information may be extracted about the lens-source system from such candidate signals by applying them to a number of O3 candidate events, even if these signals did not yield a high significance for any of the lensing hypotheses. For strongly-lensed candidates, we verify their significance using a background of simulated unlensed events and statistics computed from lensing catalogs. We also look for potential electromagnetic counterparts. In addition, we analyse in detail a candidate for a strongly-lensed sub-threshold counterpart that is identified by a new method. For microlensing candidates, we perform model selection using a number of lens models to investigate our ability to determine the mass density profile of the lens and constrain the lens parameters. We also look for millilensing signatures in one of the lensed candidates. Applying these additional analyses does not lead to any additional evidence for lensing in the candidates that have been examined. However, it does provide important insight into potential avenues to deal with high-significance candidates in future observations., Comment: 29 pages, 27 figures

We present a novel fully Bayesian analysis to constrain short gamma-ray burst jet structures associated with cocoon, wide-angle and simple top-hat jet models, as well as the binary neutron star merger rate. These constraints are made given the distance and inclination information from GW170817, observed flux of GRB170817A, observed rate of short gamma-ray bursts detected by Swift, and the neutron star merger rate inferred from LIGO's first and second observing runs. A separate analysis is conducted where a fitted short gamma-ray burst luminosity function is included to provide further constraints. The jet structure models are further constrained using the observation of GW190425 and we find that the assumption that it produced a GRB170817-like short gamma-ray burst that went undetected due to the jet geometry is consistent with previous observations. We find and quantify evidence for low luminosity and wide-angled jet structuring in the short gamma-ray burst population, independently from afterglow observations, with log Bayes factors of $0.45{-}0.55$ for such models when compared to a classical top-hat jet. Slight evidence is found for a Gaussian jet structure model over all others when the fitted luminosity function is provided, producing log Bayes factors of $0.25{-}0.9\pm0.05$ when compared to the other models. However without considering GW190425 or the fitted luminosity function, the evidence favours a cocoon-like model with log Bayes factors of $0.14\pm0.05$ over the Gaussian jet structure. We provide new constraints to the binary neutron star merger rates of $1{-}1300$Gpc$^{-3}$yr$^{-1}$ or $2{-}680$Gpc$^{-3}$yr$^{-1}$ when a fitted luminosity function is assumed., Comment: 18 pages, 8 figures

We present an improved version of the nested sampling algorithm nessai in which the core algorithm is modified to use importance weights. In the modified algorithm, samples are drawn from a mixture of normalising flows and the requirement for samples to be independently and identically distributed (i.i.d.) according to the prior is relaxed. Furthermore, it allows for samples to be added in any order, independently of a likelihood constraint, and for the evidence to be updated with batches of samples. We call the modified algorithm i-nessai. We first validate i-nessai using analytic likelihoods with known Bayesian evidences and show that the evidence estimates are unbiased in up to 32 dimensions. We compare i-nessai to standard nessai for the analytic likelihoods and the Rosenbrock likelihood, the results show that i-nessai is consistent with nessai whilst producing more precise evidence estimates. We then test i-nessai on 64 simulated gravitational-wave signals from binary black hole coalescence and show that it produces unbiased estimates of the parameters. We compare our results to those obtained using standard nessai and dynesty and find that i-nessai requires 2.68 and 13.3 times fewer likelihood evaluations to converge, respectively. We also test i-nessai of an 80-second simulated binary neutron star signal using a Reduced-Order-Quadrature (ROQ) basis and find that, on average, it converges in 24 minutes, whilst only requiring $1.01\times10^6$ likelihood evaluations compared to $1.42\times10^6$ for nessai and $4.30\times10^7$ for dynesty. These results demonstrate the i-nessai is consistent with nessai and dynesty whilst also being more efficient., Comment: 41 pages, 12 figures, 2 tables, published in Machine Learning: Science and Technology, code available at https://doi.org/10.5281/zenodo.8124198, updated with accepted version

Observations of gravitational waves (GWs) from compact binary coalescences provide powerful tests of general relativity (GR), but systematic errors in data analysis could lead to incorrect scientific conclusions. This issue is especially serious in the third-generation GW detectors in which the signal-to-noise ratio (SNR) is high and the number of detections is large. In this work, we investigate the impacts of overlapping signals and inaccurate waveform models on tests of GR. We simulate mock catalogs for Einstein Telescope and Cosmic Explorer and perform parametric tests of GR using waveform models with different levels of inaccuracy. We find the systematic error in non-GR parameter estimates could accumulate toward a false deviation from GR when combining results from multiple events, although a bayesian model selection analysis may not favour a deviation. Waveform inaccuracies contribute most to the systematic errors, but multiple overlapping signals could magnify the effects of systematics due to the incorrect removal of signals. We also point out that testing GR using selected ''golden binaries'' with high SNR is even more vulnerable to false deviations from GR. The problem of error accumulation is universal; we emphasize that it must be addressed to fully exploit the data from third-generation GW detectors, and that further investigations, particularly in waveform accuracy, will be essential., Comment: 13 pages, 6 figures

Since the first detection in 2015 of gravitational waves from compact binary coalescence, improvements to the Advanced LIGO and Advanced Virgo detectors have expanded our view into the universe for these signals. Searches of the of the latest observing run (O3) have increased the number of detected signals to 90, at a rate of approximately 1 per week. Future observing runs are expected to increase this even further. Bayesian analysis of the signals can reveal the properties of the coalescing black holes and neutron stars by comparing predicted waveforms to the observed data. The proliferating number of detected signals, the increasing number of methods that have been deployed, and the variety of waveform models create an ever-expanding number of analyses that can be considered. Asimov is a python package which is designed to simplify and standardise the process of configuring these analyses for a large number of events. It has already been used in developing analyses in three major gravitational wave catalog publications., Comment: Submitted to Journal of Open Source Software

We review Skilling's nested sampling (NS) algorithm for Bayesian inference and more broadly multi-dimensional integration. After recapitulating the principles of NS, we survey developments in implementing efficient NS algorithms in practice in high-dimensions, including methods for sampling from the so-called constrained prior. We outline the ways in which NS may be applied and describe the application of NS in three scientific fields in which the algorithm has proved to be useful: cosmology, gravitational-wave astronomy, and materials science. We close by making recommendations for best practice when using NS and by summarizing potential limitations and optimizations of NS., Comment: 20 pages + supplementary information, 5 figures. preprint version; published version at https://www.nature.com/articles/s43586-022-00121-x

With the improvement in sensitivity of gravitational wave (GW) detectors and the increasing diversity of GW sources, there is a strong need for accurate GW waveform models for data analysis. While the current model accuracy assessments require waveforms generated by numerical relativity (NR) simulations as the "true waveforms", in this paper we propose an assessment approach that does not require NR simulations, which enables us to assess model accuracy everywhere in the parameter space. By measuring the difference between two waveform models, we derive a necessary condition for a pair of waveform models to both be accurate, for a particular set of parameters. We then apply this method to the parameter estimation samples of the Gravitational-Wave Transient Catalogs GWTC-3 and GWTC-2.1, and find that the waveform accuracy for high signal-to-noise ratio events in some cases fails our assessment criterion. Based on analysis of real events' posterior samples, we discuss the correlation between our quantified accuracy assessments and systematic errors in parameter estimation. We find waveform models that perform worse in our assessment are more likely to give inconsistent estimations. We also investigate waveform accuracy in different parameter regions, and find the accuracy degrades as the spin effects go up, the mass ratio deviates from one, or the orbital plane is near-aligned to the line of sight. Furthermore, we make predictions of waveform accuracy requirements for future detectors and find the accuracy of current waveform models should be improved by at least 3 orders of magnitude, which is consistent with previous works., Comment: 17 pages, 6 figures. Accepted by PRD

The general-relativistic phenomenon of spin-induced orbital precession has not yet been observed in strong-field gravity. Gravitational-wave observations of binary black holes (BBHs) are prime candidates, since we expect the astrophysical binary population to contain precessing binaries. Imprints of precession have been investigated in several signals, but no definitive identification of orbital precession has been reported in any one of the 84 BBH observations to date by the Advanced LIGO and Virgo detectors. Here we report the measurement of strong-field precession in the LIGO-Virgo-Kagra (LVK) gravitational-wave signal GW200129. The binary's orbit precesses at a rate ten orders of magnitude faster than previous weak-field measurements from binary pulsars. We also find that the primary black hole is likely highly spinning. According to current binary population estimates a GW200129-like signal is extremely unlikely, and therefore presents a direct challenge to many current binary formation models., Comment: 16 pages, 6 figures. Update to match published version

The use of gravitational wave standard sirens for cosmological analyses is becoming well known, with particular interest in measuring the Hubble constant, $H_0$, and in shedding light on the current tension between early- and late-time measurements. The current tension is over $4\sigma$ and standard sirens will be able to provide a completely independent measurement. Dark sirens (binary black hole or neutron star mergers with no electromagnetic counterparts) can be informative if the missing redshift information is provided through the use of galaxy catalogues to identify potential host galaxies of the merger. However, galaxy catalogue incompleteness affects this analysis, and accurate modelling of it is essential for obtaining an unbiased measurement of $H_0$. Previously this has been done by assuming uniform completeness within the sky area of a gravitational wave event. This paper presents an updated methodology in which the completeness of the galaxy catalogue is estimated in a directionally-dependent matter, by pixelating the sky and computing the completeness of the galaxy catalogue along each line of sight. The $H_0$ inference for a single event is carried out on a pixel-by-pixel basis, and the pixels are combined for the final result. A reanalysis of the events in the first gravitational wave transient catalogue (GWTC-1) leads to an improvement on the measured value of $H_0$ of approximately 5% compared to the 68.3% highest-density interval of the equivalent LIGO and Virgo result, with $H_0= 68.8^{+15.9}_{-7.8}$ km s$^{-1}$ Mpc$^{-1}$., Comment: 14 pages, 10 figures

In the field of multi-messenger astronomy, Bayesian inference is commonly adopted to compare the compatibility of models given the observed data. However, to describe a physical system like neutron star mergers and their associated gamma-ray burst (GRB) events, usually more than ten physical parameters are incorporated in the model. With such a complex model, likelihood evaluation for each Monte Carlo sampling point becomes a massive task and requires a significant amount of computational power. In this work, we perform quick parameter estimation on simulated GRB X-ray light curves using an interpolated physical GRB model. This is achieved by generating a grid of GRB afterglow light curves across the parameter space and replacing the likelihood with a simple interpolation function in the high-dimensional grid that stores all light curves. This framework, compared to the original method, leads to a $\sim$90$\times$ speedup per likelihood estimation. It will allow us to explore different jet models and enable fast model comparison in the future., Comment: 9 pages, 4 figures, accepted to the special issue of Universe, "Waiting for GODOT -- Present and Future of Multi-Messenger Astronomy"

Quantum computational devices, currently under development, have the potential to accelerate data analysis techniques beyond the ability of any classical algorithm. We propose the application of a quantum algorithm for the detection of unknown signals in noisy data. We apply Grover's algorithm to matched-filtering, a signal processing technique that compares data to a number of candidate signal templates. In comparison to the classical method, this provides a speed-up proportional to the square-root of the number of templates, which would make possible otherwise intractable searches. We demonstrate both a proof-of-principle quantum circuit implementation, and a simulation of the algorithm's application to the detection of the first gravitational wave signal GW150914. We discuss the time complexity and space requirements of our algorithm as well as its implications for the currently computationally-limited searches for continuous gravitational waves., Comment: 24 pages, 19 Figures

For gravitational wave (GW) detected neutron star mergers, one of the leading candidates for electromagnetic (EM) counterparts is the afterglow from an ultra-relativistic jet. Where this afterglow is observed, it will likely be viewed off-axis, such as the afterglow following GW170817/GRB 170817A. The temporal behaviour of an off-axis observed GRB afterglow can be used to reveal the lateral jet structure, and statistical model fits can put constraints on the various model free-parameters. Amongst these parameters is the inclination of the system to the line of sight. Along with the GW detection, the afterglow modelling provides the best constraint on the inclination to the line-of-sight and can improve the estimates of cosmological parameters e.g. the Hubble constant, from GW-EM events. However, modelling of the afterglow depends on the assumed jet structure and, often overlooked, the effects of lateral spreading. Here we show how the inclusion of lateral spreading in the afterglow models can affect the estimated inclination of GW-EM events., Comment: 14 pages, 5 figures, Accepted to the special issue of Universe, "Waiting for GODOT -- Present and Future of Multi-Messenger Astronomy"

Causality and the generalized laws of black hole thermodynamics imply a bound, known as the \textit{Bekenstein--Hod universal bound}, on the information emission rate of a perturbed system. Using a time-domain ringdown analysis, we investigate whether remnant black holes produced by the coalescences observed by Advanced LIGO and Advanced Virgo obey this bound. We find that the bound is verified by the astrophysical black hole population with $94\%$ probability, providing a first confirmation of the Bekenstein--Hod bound from black hole systems., Comment: 7 pages, 2 figures, 1 Table

We present a novel method for sampling iso-likelihood contours in nested sampling using a type of machine learning algorithm known as normalising flows and incorporate it into our sampler nessai. Nessai is designed for problems where computing the likelihood is computationally expensive and therefore the cost of training a normalising flow is offset by the overall reduction in the number of likelihood evaluations. We validate our sampler on 128 simulated gravitational wave signals from compact binary coalescence and show that it produces unbiased estimates of the system parameters. Subsequently, we compare our results to those obtained with dynesty and find good agreement between the computed log-evidences whilst requiring 2.07 times fewer likelihood evaluations. We also highlight how the likelihood evaluation can be parallelised in nessai without any modifications to the algorithm. Finally, we outline diagnostics included in nessai and how these can be used to tune the sampler's settings., Comment: 20 pages, 11 figures, accepted to PRD, typos corrected, added new figure, other minor corrections, code available at https://doi.org/10.5281/zenodo.4550693

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., Comment: 15 pages, 11 figures, 1 appendix

We present a thorough observational investigation of the heuristic quantised ringdown model presented in Foit & Kleban (2019). This model is based on the Bekenstein-Mukhanov conjecture, stating that the area of a black hole horizon is an integer multiple of the Planck area $l_P^2$ multiplied by a phenomenological constant, $\alpha$, which can be viewed as an additional black hole intrinsic parameter. Our approach is based on a time-domain analysis of the gravitational wave signals produced by the ringdown phase of binary black hole mergers detected by the LIGO and Virgo collaboration. Employing a full Bayesian formalism and taking into account the complete correlation structure among the black hole parameters, we show that the value of $\alpha$ cannot be constrained using only GW150914, in contrast to what was suggested in Foit & Kleban (2019). We proceed to repeat the same analysis on the new gravitational wave events detected by the LIGO and Virgo Collaboration up to 1 October 2019, obtaining a combined-event measure equal to $\alpha = 15.6^{+20.5}_{-13.3}$ and a combined log odds ratio of $0.1 \pm 0.6$, implying that current data are not informative enough to favour or discard this model against general relativity. We then show that using a population of $\mathcal{O}(20)$ GW150914-like simulated events - detected by the current infrastructure of ground-based detectors at their design sensitivity - it is possible to confidently falsify the quantised model or prove its validity, in which case probing $\alpha$ at the few % level. Finally we classify the stealth biases that may show up in a population study., Comment: 24 pages, 8 figures. v2: updated 1 figure, version accepted by CQG (2021)

The Advanced LIGO and Advanced Virgo gravitational wave detectors have detected a population of binary black hole mergers in their first two observing runs. For each of these events we have been able to associate a potential sky location region represented as a probability distribution on the sky. Thus, at this point we may begin to ask the question of whether this distribution agrees with the isotropic model of the Universe, or if there is any evidence of anisotropy. We perform Bayesian model selection between an isotropic and a simple anisotropic model, taking into account the anisotropic selection function caused by the underlying antenna patterns and sensitivity of the interferometers over the sidereal day. We find an inconclusive Bayes factor of $1.3:1$, suggesting that the data from the first two observing runs is insufficient to pick a preferred model. However, the first detections were mostly poorly localised in the sky (before the Advanced Virgo joined the network), spanning large portions of the sky and hampering detection of potential anisotropy. It will be appropriate to repeat this analysis with events from the recent third LIGO observational run and a more sophisticated cosmological model., Comment: 8 pages, 8 figures. Accepted by MNRAS

A structured gamma-ray burst jet could explain the dimness of the prompt emission observed from GRB$\,170817$A but the exact form of this structure is still ambiguous. However, with the promise of future joint gravitational wave and gamma-ray burst observations, we shall be able to examine populations of binary neutron star mergers rather than a case-by-case basis. We present an analysis that considers gravitational wave triggered binary neutron star events both with and without short gamma-ray burst counterparts assuming that events without a counterpart were observed off-axis. This allows for Bayes factors to be calculated to compare different jet structure models. We perform model comparison between a Gaussian and power-law apparent jet structure on simulated data to demonstrate that the correct model can be distinguished with a log Bayes factor of $>5$ after less than 100 events. Constraints on the apparent structure jet model parameters are also made. After 25(100) events the angular width of the core of a power-law jet structure can be constrained within a $90\%$ credible interval of width $ \sim9.1(4.4)^{\circ} $, and the outer beaming angle to be within $\sim19.9(8.5)^{\circ}$. Similarly we show the width of a Gaussian jet structure to be constrained to $\sim2.8(1.6)^{\circ}$.

The observation of binary neutron star merger GW170817, along with its optical counterpart, provided the first constraint on the Hubble constant $H_0$ using gravitational wave standard sirens. When no counterpart is identified, a galaxy catalog can be used to provide the necessary redshift information. However, the true host might not be contained in a catalog which is not complete out to the limit of gravitational-wave detectability. These electromagnetic and gravitational-wave selection effects must be accounted for. We describe and implement a method to estimate $H_0$ using both the counterpart and the galaxy catalog standard siren methods. We perform a series of mock data analyses using binary neutron star mergers to confirm our ability to recover an unbiased estimate of $H_0$. Our simulations used a simplified universe with no redshift uncertainties or galaxy clustering, but with different magnitude-limited catalogs and assumed host galaxy properties, to test our treatment of both selection effects. We explore how the incompleteness of catalogs affects the final measurement of $H_0$, as well as the effect of weighting each galaxy's likelihood of being a host by its luminosity. In our most realistic simulation, where the simulated catalog is about three times denser than the density of galaxies in the local universe, we find that a 4.4\% measurement precision can be reached using galaxy catalogs with 50\% completeness and $\sim 250$ binary neutron star detections with sensitivity similar to that of Advanced LIGO's second observing run., Comment: 19 pages, 8 figures

The detection of the least damped quasi-normal mode from the remnant of the gravitational wave event GW150914 realised the long sought possibility to observationally study the properties of quasi-stationary black hole spacetimes through gravitational waves. Past literature has extensively explored this possibility and the emerging field has been named "black hole spectroscopy". In this study, we present results regarding the ringdown spectrum of GW150914, obtained by application of Bayesian inference to identify and characterise the ringdown modes. We employ a pure time-domain analysis method which infers from the data the time of transition between the non-linear and quasi-linear regime of the post-merger emission in concert with all other parameters characterising the source. We find that the data provides no evidence for the presence of more than one quasi-normal mode. However, from the central frequency and damping time posteriors alone, no unambiguous identification of a single mode is possible. More in-depth analysis adopting a ringdown model based on results in perturbation theory over the Kerr metric, confirms that the data do not provide enough evidence to discriminate among an $l=2$ and the $l=3$ subset of modes. Our work provides the first comprehensive agnostic framework to observationally investigate astrophysical black holes' ringdown spectra., Comment: 9 pages, 8 figures

We propose a novel methodology to select host galaxy candidates of future pulsar timing array (PTA) detections of resolved gravitational waves (GWs) from massive black hole binaries (MBHBs). The method exploits the physical dependence of the GW amplitude on the MBHB chirp mass and distance to the observer, together with empirical MBH mass-host galaxy correlations, to rank potential host galaxies in the mass-redshift plane. This is coupled to a null-stream based likelihood evaluation of the GW amplitude and sky position in a Bayesian framework that assigns to each galaxy a probability of hosting the MBHB generating the GW signal. We test our algorithm on a set of realistic simulations coupling the likely properties of the first PTA resolved GW signal to synthetic all-sky galaxy maps. For a foreseeable PTA sky-localization precision of 100 squared degrees, we find that the GW source is hosted with 50% (90%) probability within a restricted number of <50 (<500) potential hosts. These figures are orders of magnitude smaller than the total number of galaxies within the PTA sky error-box, enabling extensive electromagnetic follow-up campaigns on a limited number of targets., Comment: 12 pages, 10 figures

Advanced LIGO and Virgo have so far detected gravitational waves from 10 binary black hole mergers (BBH) and 1 binary neutron star merger (BNS). In the future, we expect the detection of many more marginal sources, since compact binary coalescences detectable by advanced ground-based instruments are roughly distributed uniformly in comoving volume. In this paper we simulate weak signals from compact binary coalescences of various morphologies and optimal network signal-to-noise ratios (henceforth SNRs), and analyze if and to which extent their parameters can be measured by advanced LIGO and Virgo in their third observing run. We show that subthreshold binary neutron stars, with SNRs below 12 (10) yield uncertainties in their sky position larger than 400 (700) $\mathrm{deg}^2$ (90% credible interval). The luminosity distance, which could be used to measure the Hubble constant with standard sirens, has relative uncertainties larger than 40% for BNSs and neutron star black hole mergers. For sources with SNRs below 8, it is not uncommon that the extrinsic parameters, sky position and distance, cannot be measured. Next, we look at the intrinsic parameters, masses and spins. We show that the detector-frame chirp mass can sometimes be measured with uncertainties below 1% even for sources at SNRs of 6, although multimodality is not uncommon and can significantly broaden the posteriors. The effective inspiral spin is best measured for neutron star black hole mergers, for which the uncertainties can be as low as $\sim0.08$ ($\sim 0.2$) at SNR 12 (8). The uncertainty is higher for systems with comparable component masses or lack of spin precession., Comment: 15 pages, 12 figures, 6 tables; Results unchanged from previous versions, provided additional details; public data release for posterior samples: https://git.ligo.org/yiwen.huang/public-lowsnr-data-release/

Bayesian statistical inference has become increasingly important for the analysis of observations from the Advanced LIGO and Advanced Virgo gravitational-wave detectors. To this end, iterative simulation techniques, in particular nested sampling and parallel tempering, have been implemented in the software library LALInference to sample from the posterior distribution of waveform parameters of compact binary coalescence events. Nested sampling was mainly developed to calculate the marginal likelihood of a model but can produce posterior samples as a by-product. Thermodynamic integration is employed to calculate the evidence using samples generated by parallel tempering but has been found to be computationally demanding. Here we propose the stepping-stone sampling algorithm, originally proposed by Xie et al. (2011) in phylogenetics and a special case of path sampling, as an alternative to thermodynamic integration. The stepping-stone sampling algorithm is also based on samples from the power posteriors of parallel tempering but has superior performance as fewer temperature steps and thus computational resources are needed to achieve the same accuracy. We demonstrate its performance and computational costs in comparison to thermodynamic integration and nested sampling in a simulation study and a case study of computing the marginal likelihood of a binary black hole signal model applied to simulated data from the Advanced LIGO and Advanced Virgo gravitational wave detectors. To deal with the inadequate methods currently employed to estimate the standard errors of evidence estimates based on power posterior techniques, we propose a novel block bootstrap approach and show its potential in our simulation study and LIGO application., Comment: 10 pages, 5 figures, 2 tables

Coalescing compact binaries emitting gravitational wave (GW) signals, as recently detected by the Advanced LIGO-Virgo network, constitute a population over the multi-dimensional space of component masses and spins, redshift, and other parameters. Characterizing this population is a major goal of GW observations and may be approached via parametric models. We demonstrate hierarchical inference for such models with a method that accounts for uncertainties in each binary merger's individual parameters, for mass-dependent selection effects, and also for the presence of a second population of candidate events caused by detector noise. Thus, the method is robust to potential biases from a contaminated sample and allows us to extract information from events that have a relatively small probability of astrophysical origin., Comment: 19 pages, 12 figures

Thanks to the recent discoveries of gravitational wave signals from binary black hole mergers by Advanced Laser Interferometer Gravitational Wave Observatory and Advanced Virgo, the genuinely strong-field dynamics of spacetime can now be probed, allowing for stringent tests of general relativity (GR). One set of tests consists of allowing for parametrized deformations away from GR in the template waveform models and then constraining the size of the deviations, as was done for the detected signals in previous work. In this paper, we construct reduced-order quadratures so as to speed up likelihood calculations for parameter estimation on future events. Next, we explicitly demonstrate the robustness of the parametrized tests by showing that they will correctly indicate consistency with GR if the theory is valid. We also check to what extent deviations from GR can be constrained as information from an increasing number of detections is combined. Finally, we evaluate the sensitivity of the method to possible violations of GR., Comment: 19 pages, many figures. Matches PRD version

Super-massive black hole binaries are expected to produce a GW signal in the nano-Hertz frequency band which may be detected by PTAs in the coming years. The signal is composed of both stochastic and individually resolvable components. Here we develop a generic Bayesian method for the analysis of resolvable sources based on the construction of `null-streams' which cancel the part of the signal held in common for each pulsar (the Earth-term). For an array of $N$ pulsars there are $N-2$ independent null-streams that cancel the GW signal from a particular sky location. This method is applied to the localisation of quasi-circular binaries undergoing adiabatic inspiral. We carry out a systematic investigation of the scaling of the localisation accuracy with signal strength and number of pulsars in the PTA. Additionally, we find that source sky localisation with the International PTA data release one is vastly superior than what is achieved by its constituent regional PTAs., Comment: 13 pages, 7 figures, 1 appendix. Edited Figures 5, 6, 7 due to a bug in the plotting script (results unchanged). Additional edit to fix a type in equation 4

The statistical significance of a candidate gravitational-wave (GW) event is crucial to the prospects for a confirmed detection, or for its selection as a candidate for follow-up electromagnetic observation. To determine the significance of a GW candidate, a ranking statistic is evaluated and compared to an empirically-estimated background distribution, yielding a false alarm probability or p-value. The reliability of this background estimate is limited by the number of background samples and by the fact that GW detectors cannot be shielded from signals, making it impossible to identify a pure background data set. Different strategies have been proposed: in one method, all samples, including potential signals, are included in the background estimation, whereas in another method, coincidence removal is performed in order to exclude possible signals from the estimated background. Here we report on a mock data challenge, performed prior to the first detections of GW signals by Advanced LIGO, to compare these two methods. The all-samples method is found to be self-consistent in terms of the rate of false positive detection claims, but its p-value estimates are systematically conservative and subject to higher variance. Conversely, the coincidence-removal method yields a mean-unbiased estimate of the p-value but sacrifices self-consistency. We provide a simple formula for the uncertainty in estimate significance and compare it to mock data results. Finally, we discuss the use of different methods in claiming the detection of GW signals., Comment: A shortened version of arXiv:1601.00130

Motivated by the preponderance of so-called "heavy black holes" in the binary black hole (BBH) gravitational wave (GW) detections to date, and the role that gravitational lensing continues to play in discovering new galaxy populations, we explore the possibility that the GWs are strongly-lensed by massive galaxy clusters. For example, if one of the GW sources were actually located at $z=1$, then the rest-frame mass of the associated BHs would be reduced by a factor $\sim2$. Based on the known populations of BBH GW sources and strong-lensing clusters, we estimate a conservative lower limit on the number of BBH mergers detected per detector year at LIGO/Virgo's current sensitivity that are multiply-imaged, of $R_{\rm detect}\simeq10^{-5}{\rm yr}^{-1}$. This is equivalent to rejecting the hypothesis that one of the BBH GWs detected to date was multiply-imaged at $<\sim4\sigma$. It is therefore unlikely but not impossible that one of the GWs is multiply-imaged. We identify three spectroscopically confirmed strong-lensing clusters with well constrained mass models within the $90\%$ credible sky localisations of the BBH GWs from LIGO's first observing run. In the event that one of these clusters multiply-imaged one of the BBH GWs, we predict that $20-60\%$ of the putative next appearances of the GWs would be detectable by LIGO, and that they would arrive at Earth within three years of first detection., Comment: 7 pages, 3 figures, accepted by MNRAS

This document describes a code to perform parameter estimation and model selection in targeted searches for continuous gravitational waves from known pulsars using data from ground-based gravitational wave detectors. We describe the general workings of the code and characterise it on simulated data containing both noise and simulated signals. We also show how it performs compared to a previous MCMC and grid-based approach to signal parameter estimation. Details how to run the code in a variety of cases are provided in Appendix A., Comment: 46 pages, 34 figures. Document detailing and characterising analysis software for a gravitational wave search. The document source, example scripts and jupyter notebooks used can be found at https://github.com/mattpitkin/CW_nested_sampling_doc

The first detected gravitational wave signal, GW150914, was produced by the coalescence of a stellar-mass binary black hole. Along with the subsequent detection of GW151226, GW170104 and the candidate event LVT151012, this gives us evidence for a population of black hole binaries with component masses in the tens of solar masses. As detector sensitivity improves, this type of source is expected to make a large contribution to the overall number of detections, but has received little attention compared to binary neutron star systems in studies of projected network performance. We simulate the observation of a system like GW150914 with different proposed network configurations, and study the precision of parameter estimates, particularly source location, orientation and masses. We find that the improvements to low frequency sensitivity that are expected with continued commissioning will improve the precision of chirp mass estimates by an order of magnitude, whereas the improvements in sky location and orientation are driven by the expanded network configuration. This demonstrates that both sensitivity and number of detectors will be important factors in the scientific potential of second generation detector networks., Comment: 18 pages, 5 figures, 2 tables

Parameter estimation on gravitational wave signals from compact binary coalescence (CBC) requires the evaluation of computationally intensive waveform models, typically the bottleneck in the analysis. This cost will increase further as low frequency sensitivity in later second and third generation detectors motivates the use of longer waveforms. We describe a method for accelerating parameter estimation by exploiting the chirping behaviour of the signals to sample the waveform sparsely for portions where the full frequency resolution is not required. We demonstrate that the method can reproduce the original results with a waveform mismatch of $\leq 5\times 10^{-7}$, but with a waveform generation cost up to $\sim 50$ times lower for computationally costly frequency-domain waveforms starting from below 8 Hz.

The era of gravitational-wave astronomy has started with the discovery of the binary black hole coalescences (BBH) GW150914 and GW151226 by the LIGO instruments. These systems allowed for the first direct measurement of masses and spins of black holes. The component masses in each of the systems have been estimated with uncertainties of over 10\%, with only weak constraints on the spin magnitude and orientation. In this paper we show how these uncertainties will be typical for this type of source when using advanced detectors. Focusing in particular on heavy BBH of masses similar to GW150914, we find that typical uncertainties in the estimation of the source-frame component masses will be around 40\%. We also find that for most events the magnitude of the component spins will be estimated poorly: for only 10\% of the systems the uncertainties in the spin magnitude of the primary (secondary) BH will be below 0.7 (0.8). Conversely, the effective spin along the angular momentum can be estimated more precisely than either spins, with uncertainties below 0.16 for 10\% of the systems. We also quantify how often large or negligible primary spins can be excluded, and how often the sign of the effective spin can be measured. We show how the angle between the spin and the orbital angular momentum can only seldom be measured with uncertainties below 60$^\circ$. We then investigate how the measurement of spin parameters depends on the inclination angle and the total mass of the source. We find that when precession is present, uncertainties are smaller for systems observed close to edge-on. Contrarily to what happens for low-mass, inspiral dominated, sources, for heavy BBH we find that large spins aligned with the orbital angular momentum can be measured with small uncertainty. We also show how spin uncertainties increase with the total mass. Finally..., Comment: 18 pages, 28 figures. The abstract is cut in the Arxiv metadata. Refer to PDF. Version accepted by PRD

The recent advanced LIGO detections of gravitational waves from merging binary black holes enhance the prospect of exploring binary evolution via gravitational-wave observations of a population of compact-object binaries. In the face of uncertainty about binary formation models, model-independent inference provides an appealing alternative to comparisons between observed and modelled populations. We describe a procedure for clustering in the multi-dimensional parameter space of observations that are subject to significant measurement errors. We apply this procedure to a mock data set of population-synthesis predictions for the masses of merging compact binaries convolved with realistic measurement uncertainties, and demonstrate that we can accurately distinguish subpopulations of binary neutron stars, binary black holes, and mixed neutron star -- black hole binaries with tens of observations., Comment: Replacing with published version; minor typo fixes and reference updates

We investigate the issue in determining the significance of candidate transient gravitational-wave events in a ground-based interferometer network. Given the presence of non-Gaussian noise artefacts in real data, the noise background must be estimated empirically from the data itself. However, the data also potentially contains signals, thus the background estimate may be overstated due to contributions from signals. It has been proposed to mitigate possible bias by removing single-detector data samples that pass a multi-detector consistency test from the background estimates. We conduct a high-statistics Mock Data Challenge to evaluate the effects of removing such samples, modelling a range of scenarios with plausible detector noise distributions and with a range of plausible foreground astrophysical signal rates. We consider the two different modes: one in which coincident samples are removed, and one in which all samples are retained and used. Three algorithms were operated in both modes, show good consistency with each other; however, discrepancies arise between the results obtained under the "coincidence removal" and "all samples" modes, for false alarm probabilities below a certain value. In most scenarios the median of the false alarm probability (FAP) estimator under the "all samples" mode is consistent with the exact FAP. On the other hand the "coincidence removal" mode is found to be unbiased for the mean of the estimated FAP. While the numerical values at which discrepancies become apparent are specific to the details of our experiment, we believe that the qualitative differences in the behaviour of the median and mean of the FAP estimator have more general validity. On the basis of our study we suggest that the FAP of candidates for the first detection of gravitational waves should be estimated without removing single-detector samples that form coincidences., Comment: 30 pages

Gravitational waves from coalescences of neutron stars or stellar-mass black holes into intermediate-mass black holes (IMBHs) of $\gtrsim 100$ solar masses represent one of the exciting possible sources for advanced gravitational-wave detectors. These sources can provide definitive evidence for the existence of IMBHs, probe globular-cluster dynamics, and potentially serve as tests of general relativity. We analyse the accuracy with which we can measure the masses and spins of the IMBH and its companion in intermediate-mass ratio coalescences. We find that we can identify an IMBH with a mass above $100 ~ M_\odot$ with $95\%$ confidence provided the massive body exceeds $130 ~ M_\odot$. For source masses above $\sim200 ~ M_\odot$, the best measured parameter is the frequency of the quasi-normal ringdown. Consequently, the total mass is measured better than the chirp mass for massive binaries, but the total mass is still partly degenerate with spin, which cannot be accurately measured. Low-frequency detector sensitivity is particularly important for massive sources, since sensitivity to the inspiral phase is critical for measuring the mass of the stellar-mass companion. We show that we can accurately infer source parameters for cosmologically redshifted signals by applying appropriate corrections. We investigate the impact of uncertainty in the model gravitational waveforms and conclude that our main results are likely robust to systematics., Comment: 9 pages, 11 figures

Inspiraling binary neutron stars are expected to be one of the most significant sources of gravitational-wave signals for the new generation of advanced ground-based detectors. We investigate how well we could hope to measure properties of these binaries using the Advanced LIGO detectors, which began operation in September 2015. We study an astrophysically motivated population of sources (binary components with masses $1.2~\mathrm{M}_\odot$--$1.6~\mathrm{M}_\odot$ and spins of less than $0.05$) using the full LIGO analysis pipeline. While this simulated population covers the observed range of potential binary neutron-star sources, we do not exclude the possibility of sources with parameters outside these ranges; given the existing uncertainty in distributions of mass and spin, it is critical that analyses account for the full range of possible mass and spin configurations. We find that conservative prior assumptions on neutron-star mass and spin lead to average fractional uncertainties in component masses of $\sim 16\%$, with little constraint on spins (the median $90\%$ upper limit on the spin of the more massive component is $\sim 0.7$). Stronger prior constraints on neutron-star spins can further constrain mass estimates, but only marginally. However, we find that the sky position and luminosity distance for these sources are not influenced by the inclusion of spin; therefore, if LIGO detects a low-spin population of BNS sources, less computationally expensive results calculated neglecting spin will be sufficient for guiding electromagnetic follow-up., Comment: 10 pages, 9 figures

In the fall of 2015 the first scientific observing run (O1) of the advanced LIGO detectors will be conducted. Based on the recent commissioning progress at the LIGO Hanford and Livingston sites, the gravitational wave detector range for a neutron star binary inspiral is expected to be of order 60 Mpc. We describe here our planning for an O1 search for coincidence between a LIGO gravitational wave detection and a gamma-ray signal from the Fermi Gamma-ray Burst Monitor. Such a coincidence would constitute measurement of an electromagnetic counterpart to a gravitational wave signal, with significant corresponding scientific benefits, including revealing the central engine powering the gamma-ray burst, enhanced confidence in the event as a genuine astrophysical detection, and a determination of the relative speed of the photon and graviton., Comment: Submitted to Moriond Gravitation Conference Proceedings 2015

We perform a systematic study to explore the accuracy with which the parameters of intermediate-mass black-hole binary systems can be measured from their gravitational wave (GW) signatures using second-generation GW detectors. We make use of the most recent reduced-order models containing inspiral, merger and ringdown signals of aligned-spin effective-one-body waveforms (SEOBNR) to significantly speed up the calculations. We explore the phenomenology of the measurement accuracies for binaries with total masses between 50 and 500 $M_\odot$ and mass ratios between 0.1 and 1. We find that (i) at total masses below ~200 $M_\odot$, where the signal-to-noise-ratio is dominated by the inspiral portion of the signal, the chirp mass parameter can be accurately measured; (ii) at higher masses, the information content is dominated by the ringdown, and total mass is measured more accurately; (iii) the mass of the lower-mass companion is poorly estimated, especially at high total mass and more extreme mass ratios; (iv) spin cannot be accurately measured for our injection set with non-spinning components. Most importantly, we find that for binaries with non-spinning components at all values of the mass ratio in the considered range and at network signal-to-noise ratio of 15, analyzed with spin-aligned templates, the presence of an intermediate-mass black hole with mass >100 $M_\odot$ can be confirmed with 95% confidence in any binary that includes a component with a mass of 130 $M_\odot$ or greater., Comment: 6 pages, 8 figures; published version

Recently exploratory studies were performed on the possibility of constraining the neutron star equation of state (EOS) using signals from coalescing binary neutron stars, or neutron star-black hole systems, as they will be seen in upcoming advanced gravitational wave detectors such as Advanced LIGO and Advanced Virgo. In particular, it was estimated to what extent the combined information from multiple detections would enable one to distinguish between different equations of state through hypothesis ranking or parameter estimation. Under the assumption of zero neutron star spins both in signals and in template waveforms and considering tidal effects to 1 post-Newtonian (1PN) order, it was found that O(20) sources would suffice to distinguish between a hard, moderate, and soft equation of state. Here we revisit these results, this time including neutron star tidal effects to the highest order currently known, termination of gravitational waveforms at the contact frequency, neutron star spins, and the resulting quadrupole-monopole interaction. We also take the masses of neutron stars in simulated sources to be distributed according to a relatively strongly peaked Gaussian, as hinted at by observations, but without assuming that the data analyst will necessarily have accurate knowledge of this distribution for use as a mass prior. We find that especially the effect of the latter is dramatic, necessitating many more detections to distinguish between different EOS and causing systematic biases in parameter estimation, on top of biases due to imperfect understanding of the signal model pointed out in earlier work. This would get mitigated if reliable prior information about the mass distribution could be folded into the analyses., Comment: 18 pages, 12 figures, revised version accepted for publication in Phys. Rev. D

Advanced ground-based gravitational-wave (GW) detectors begin operation imminently. Their intended goal is not only to make the first direct detection of GWs, but also to make inferences about the source systems. Binary neutron-star mergers are among the most promising sources. We investigate the performance of the parameter-estimation \edit{(PE)} pipeline that will be used during the first observing run of the Advanced Laser Interferometer Gravitational-wave Observatory (aLIGO) in 2015: we concentrate on the ability to reconstruct the source location on the sky, but also consider the ability to measure masses and the distance. Accurate, rapid sky-localization is necessary to alert electromagnetic (EM) observatories so that they can perform follow-up searches for counterpart transient events. We consider PE accuracy in the presence of \edit{non-stationary}, non-Gaussian noise. We find that the character of the noise makes negligible difference to the PE performance \edit{at a given signal-to-noise ratio}. The source luminosity distance can only be poorly constrained, the median $90\%$ ($50\%$) credible interval scaled with respect to the true distance is $0.85$ ($0.38$). However, the chirp mass is well measured. Our chirp-mass estimates are subject to systematic error because we used gravitational-waveform templates without component spin to carry out inference on signals with moderate spins, but the total error is typically less than $10^{-3} M_\odot$. The median $90\%$ ($50\%$) credible region for sky localization is $\sim600~\mathrm{deg^{2}}$ ($\sim150~\mathrm{deg^{2}}$), with $3\%$ ($30\%$) of detected events localized within $100~\mathrm{deg^{2}}$. Early aLIGO, with only two detectors, will have a sky-localization accuracy for binary neutron stars of hundreds of square degrees; this makes EM follow-up challenging, but not impossible., Comment: 21 pages, 24 figures. Matches published version. Data release at http://www.ligo.org/scientists/first2years/

We present two different search methods for electromagnetic counterparts to gravitational-wave (GW) events from ground-based detectors using archival NASA high-energy data from the Fermi-GBM and RXTE-ASM instruments. To demonstrate the methods, we use a limited number of representative GW background noise events produced by a search for binary neutron star coalescence over the last two months of the LIGO-Virgo S6/VSR3 joint science run. Time and sky location provided by the GW data trigger a targeted search in the high-energy photon data. We use two custom pipelines: one to search for prompt gamma-ray counterparts in GBM, and the other to search for a variety of X-ray afterglow model signals in ASM. We measure the efficiency of the joint pipelines to weak gamma-ray burst counterparts, and a family of model X-ray afterglows. By requiring a detectable signal in either electromagnetic instrument coincident with a GW event, we are able to reject a large majority of GW candidates. This reduces the signal-to-noise of the loudest surviving GW background event by around 15-20%., Comment: Accepted for publication in ApJS

The Advanced LIGO and Advanced Virgo gravitational wave (GW) detectors will begin operation in the coming years, with compact binary coalescence events a likely source for the first detections. The gravitational waveforms emitted directly encode information about the sources, including the masses and spins of the compact objects. Recovering the physical parameters of the sources from the GW observations is a key analysis task. This work describes the LALInference software library for Bayesian parameter estimation of compact binary signals, which builds on several previous methods to provide a well-tested toolkit which has already been used for several studies. We show that our implementation is able to correctly recover the parameters of compact binary signals from simulated data from the advanced GW detectors. We demonstrate this with a detailed comparison on three compact binary systems: a binary neutron star, a neutron star black hole binary and a binary black hole, where we show a cross-comparison of results obtained using three independent sampling algorithms. These systems were analysed with non-spinning, aligned spin and generic spin configurations respectively, showing that consistent results can be obtained even with the full 15-dimensional parameter space of the generic spin configurations. We also demonstrate statistically that the Bayesian credible intervals we recover correspond to frequentist confidence intervals under correct prior assumptions by analysing a set of 100 signals drawn from the prior. We discuss the computational cost of these algorithms, and describe the general and problem-specific sampling techniques we have used to improve the efficiency of sampling the compact binary coalescence parameter space.

We anticipate the first direct detections of gravitational waves (GWs) with Advanced LIGO and Virgo later this decade. Though this groundbreaking technical achievement will be its own reward, a still greater prize could be observations of compact binary mergers in both gravitational and electromagnetic channels simultaneously. During Advanced LIGO and Virgo's first two years of operation, 2015 through 2016, we expect the global GW detector array to improve in sensitivity and livetime and expand from two to three detectors. We model the detection rate and the sky localization accuracy for binary neutron star (BNS) mergers across this transition. We have analyzed a large, astrophysically motivated source population using real-time detection and sky localization codes and higher-latency parameter estimation codes that have been expressly built for operation in the Advanced LIGO/Virgo era. We show that for most BNS events the rapid sky localization, available about a minute after a detection, is as accurate as the full parameter estimation. We demonstrate that Advanced Virgo will play an important role in sky localization, even though it is anticipated to come online with only one-third as much sensitivity as the Advanced LIGO detectors. We find that the median 90% confidence region shrinks from ~500 square degrees in 2015 to ~200 square degrees in 2016. A few distinct scenarios for the first LIGO/Virgo detections emerge from our simulations., Comment: 17 pages, 11 figures, 5 tables. For accompanying data, see http://www.ligo.org/scientists/first2years

Compact binary coalescences are the most promising sources of gravitational waves (GWs) for ground based detectors. Binary systems containing one or two spinning black holes are particularly interesting due to spin-orbit (and eventual spin-spin) interactions, and the opportunity of measuring spins directly through GW observations. In this letter we analyze simulated signals emitted by spinning binaries with several values of masses, spins, orientation, and signal-to-noise ratio. We find that spin magnitudes and tilt angles can be estimated to accuracy of a few percent for neutron star--black hole systems and $\sim$ 5-30% for black hole binaries. In contrast, the difference in the azimuth angles of the spins, which may be used to check if spins are locked into resonant configurations, cannot be constrained. We observe that the best performances are obtained when the line of sight is perpendicular to the system's total angular momentum, and that a sudden change of behavior occurs when a system is observed from angles such that the plane of the orbit can be seen both from above and below during the time the signal is in band. This study suggests that the measurement of black hole spin by means of GWs can be as precise as what can be obtained from X-ray binaries., Comment: 4 figures, Version accepted for publication on PRL