GW150914: First results from the search for binary black hole coalescence with Advanced LIGO

Nonlinearities in Black Hole Ringdowns

The gravitational wave strain emitted by a perturbed black hole (BH) ringing down is typically modeled analytically using first-order BH perturbation theory. In this Letter, we show that second-order effects are necessary for modeling ringdowns from BH merger simulations. Focusing on the strain's (ℓ,m)=(4,4) angular harmonic, we show the presence of a quadratic effect across a range of binary BH mass ratios that agrees with theoretical expectations. We find that the quadratic (4,4) mode's amplitude exhibits quadratic scaling with the fundamental (2,2) mode-its parent mode. The nonlinear mode's amplitude is comparable to or even larger than that of the linear (4,4) mode. Therefore, correctly modeling the ringdown of higher harmonics-improving mode mismatches by up to 2 orders of magnitude-requires the inclusion of nonlinear effects.

Compact Binary Coalescences: Astrophysical Processes and Lessons Learned

On 11 February 2016, the LIGO and Virgo scientific collaborations announced the first direct detection of gravitational waves, a signal caught by the LIGO interferometers on 14 September 2015, and produced by the coalescence of two stellar-mass black holes. The discovery represented the beginning of an entirely new way to investigate the Universe. The latest gravitational-wave catalog by LIGO, Virgo and KAGRA brings the total number of gravitational-wave events to 90, and the count is expected to significantly increase in the next years, when additional ground-based and space-born interferometers will be operational. From the theoretical point of view, we have only fuzzy ideas about where the detected events came from, and the answers to most of the five Ws and How for the astrophysics of compact binary coalescences are still unknown. In this work, we review our current knowledge and uncertainties on the astrophysical processes behind merging compact-object binaries. Furthermore, we discuss the astrophysical lessons learned through the latest gravitational-wave detections, paying specific attention to the theoretical challenges coming from exceptional events (e.g., GW190521 and GW190814).

Model-free detection of unique events in time series

Recognition of anomalous events is a challenging but critical task in many scientific and industrial fields, especially when the properties of anomalies are unknown. In this paper, we introduce a new anomaly concept called "unicorn" or unique event and present a new, model-free, unsupervised detection algorithm to detect unicorns. The key component of the new algorithm is the Temporal Outlier Factor (TOF) to measure the uniqueness of events in continuous data sets from dynamic systems. The concept of unique events differs significantly from traditional outliers in many aspects: while repetitive outliers are no longer unique events, a unique event is not necessarily an outlier; it does not necessarily fall out from the distribution of normal activity. The performance of our algorithm was examined in recognizing unique events on different types of simulated data sets with anomalies and it was compared with the Local Outlier Factor (LOF) and discord discovery algorithms. TOF had superior performance compared to LOF and discord detection algorithms even in recognizing traditional outliers and it also detected unique events that those did not. The benefits of the unicorn concept and the new detection method were illustrated by example data sets from very different scientific fields. Our algorithm successfully retrieved unique events in those cases where they were already known such as the gravitational waves of a binary black hole merger on LIGO detector data and the signs of respiratory failure on ECG data series. Furthermore, unique events were found on the LIBOR data set of the last 30 years.

Testing General Relativity with Gravitational Waves: An Overview

The detections of gravitational-wave (GW) signals from compact binary coalescence by ground-based detectors have opened up the era of GW astronomy. These observations provide opportunities to test Einstein’s general theory of relativity at the strong-field regime. Here we give a brief overview of the various GW-based tests of General Relativity (GR) performed by the LIGO-Virgo collaboration on the detected GW events to date. After providing details for the tests performed in four categories, we discuss the prospects for each test in the context of future GW detectors. The four categories of tests include the consistency tests, parametrized tests for GW generation and propagation, tests for the merger remnant properties, and GW polarization tests.

Anomaly detection in gravitational waves data using convolutional autoencoders

As of this moment, 50 gravitational wave (GW) detections have been announced, thanks to the observational efforts of the LIGO-Virgo collaboration, working with the Advanced LIGO and the Advanced Virgo interferometers. The detection of signals is complicated by the noise-dominated nature of the data. Conventional approaches in GW detection procedures require either precise knowledge of the GW waveform in the context of matched filtering searches or coincident analysis of data from multiple detectors. Furthermore, the analysis is prone to contamination by instrumental or environmental artifacts called glitches which either mimic astrophysical signals or reduce the overall quality of data. In this paper, we propose an alternative generic method of studying GW data based on detecting anomalies. The anomalies we study are transient signals, different from the slow non-stationary noise of the detector. The anomalies presented in the manuscript are mostly based on the GW emitted by the mergers of binary black hole systems. However, the presented study of anomalies is not limited only to GW alone, but also includes glitches occurring in the real LIGO/Virgo dataset available at the Gravitational Waves Open Science Center. To search for anomalies we employ deep learning algorithms, namely convolutional autoencoders, which are trained on both simulated and real detector data. We demonstrate the capabilities of our deep learning implementation in the reconstruction of injected signals. We study the influence of the GW strength, defined in terms of matched filter signal-to-noise ratio, on the detection of anomalies. Moreover, we present the application of our method for the localization in time of anomalies in the studied time-series data. We validate the results of anomaly searches on real data containing confirmed gravitational wave detections; we thus prove the generalization capabilities of our method, towards detecting GWs unknown to our deep learning models during training.

A macroscopic object passively cooled into its quantum ground state of motion beyond single-mode cooling

The nature of the quantum-to-classical crossover remains one of the most challenging open question of Science to date. In this respect, moving objects play a specific role. Pioneering experiments over the last few years have begun exploring quantum behaviour of micron-sized mechanical systems, either by passively cooling single GHz modes, or by adapting laser cooling techniques developed in atomic physics to cool specific low-frequency modes far below the temperature of their surroundings. Here instead we describe a very different approach, passive cooling of a whole micromechanical system down to 500 μK, reducing the average number of quanta in the fundamental vibrational mode at 15 MHz to just 0.3 (with even lower values expected for higher harmonics); the challenge being to be still able to detect the motion without disturbing the system noticeably. With such an approach higher harmonics and the surrounding environment are also cooled, leading to potentially much longer mechanical coherence times, and enabling experiments questioning mechanical wave-function collapse, potentially from the gravitational background, and quantum thermodynamics. Beyond the average behaviour, here we also report on the fluctuations of the fundamental vibrational mode of the device in-equilibrium with the cryostat. These reveal a surprisingly complex interplay with the local environment and allow characteristics of two distinct thermodynamic baths to be probed.

Compared to active techniques, passive cooling of mechanical modes allows to work with devices in equilibrium with their environment without excess damping. Here, the authors demonstrate passive cooling and thermalisation of a 15 μm drum-head device with MHz fundamental flexure to its quantum ground state.

A template-free approach for waveform extraction of gravitational wave events

We develop a general data-driven and template-free method for the extraction of event waveforms in the presence of background noise. Recent gravitational-wave observations provide one of the significant scientific areas requiring data analysis and waveform extraction capability. We use our method to find the waveforms for the reported events from the first, second, and third LIGO observation runs (O1, O2, and O3). Using the instantaneous frequencies derived by the Hilbert transform of the extracted waveforms, we provide the physical time delays between the arrivals of gravitational waves to the detectors.

Drop tower tests of Taiji-1 inertial sensor substitute

Taiji-1, which is the first technical verification satellite of China's Space Gravitational Wave Detection Program, was successfully launched on August 31, 2019. The mission aimed to investigate the key technologies used in space gravitational wave detection. The inertial sensor, which was one of the main payloads, measured the residual acceleration of the satellite, and verified the drag-free control technology. Its performance was crucial to the success of the Taiji-1 mission. To ensure its performance in orbit, the inertial sensor was fully evaluated prior to launch. Owing to the gravitational acceleration on the ground, it is impossible to verify all the properties of the inertial sensor in a routine laboratory. A feasible method to conduct such tests is to use a drop tower. To guarantee the safety of the inertial sensor, a substitute was used with similar structure and circuit design. A total of 20 falls in three groups were completed, a set of research methods was established, and the importance of conducting simulations before the drop tests was verified. For the first time, the switch of different circuit gains in a drop tower test has been achieved and the National Microgravity Laboratory of China (NMLC) drop tower's residual accelerations in three dimensions were measured. The results demonstrated that the microgravity level of the drop tower can reach about 58 μg₀ in the fall direction and 13 μg₀ along the horizontal axes.

Point defects in IBS coating for very low loss mirrors

High-reflective coatings are used in many physics experiments. Despite the high quality of the optical coating, the performances of the mirrors are altered by the scattered light induced by micrometer size defects in the coating layers. The topic of this paper is the study of the point-like scatterers present in the specific coating of the mirrors used in state-of-the-art, high-sensitivity optical experiments. We studied the behavior of the materials according to different thicknesses and the resulting defects after annealing. To the best of our knowledge, this is the first insight into the formation of such defects for different materials and thickness and how this is reduced when samples are annealed.

Cosmological Advection Flows in the Presence of Primordial Black Holes as Dark Matter and Formation of First Sources

In the inflation-based cosmology the dark matter (DM) density component starts moving with respect to the universal expansion at z_{eq}∼3200 while baryons remain frozen until z_{rec}∼1100. It has been suggested that in this case postlinear corrections to the evolution of small fluctuations would result, for the standard Λ-dominated cold DM (CDM) model, in delayed formation of early objects as supersonic advection flows develop after recombination, so baryons are not immediately captured by the DM gravity on small scales. We develop the hydrodynamical description of such two-component advection and show that, in the supersonic regime, the advection within irrotational fluids is governed by the gradient of the difference of the kinetic energies of the two (DM and baryonic here) components. We then apply this formalism to the case where DM is made up of LIGO-type black holes (BHs) and show that there the advection process on scales relevant for early structure collapse will differ significantly from the earlier discussed (CDM) case because of the additional granulation component to the density field produced during inflation. The advection here will lead efficiently to the common motion of the DM and baryon components on scales relevant for collapse and formation of first luminous sources. This leads to early collapse, making it easier to explain the existence of supermassive BHs observed in quasars at high z>7. The resultant net advection rate reaches minimum around ≲10^{9}  M_{⊙} and subsequently rises to a secondary maximum near the typical mass of ∼10^{12}  M_{⊙}, which may be an important consideration for formation of galaxies at z≲(a few).

Search for topological defect dark matter with a global network of optical magnetometers

Ultralight bosons such as axion-like particles are viable candidates for dark matter. They can form stable, macroscopic field configurations in the form of topological defects that could concentrate the dark matter density into many distinct, compact spatial regions that are small compared with the Galaxy but much larger than the Earth. Here we report the results of the search for transient signals from the domain walls of axion-like particles by using the global network of optical magnetometers for exotic (GNOME) physics searches. We search the data, consisting of correlated measurements from optical atomic magnetometers located in laboratories all over the world, for patterns of signals propagating through the network consistent with domain walls. The analysis of these data from a continuous month-long operation of GNOME finds no statistically significant signals, thus placing experimental constraints on such dark matter scenarios.

Einstein’s Geometrical versus Feynman’s Quantum-Field Approaches to Gravity Physics: Testing by Modern Multimessenger Astronomy

Modern multimessenger astronomy delivers unique opportunity for performing crucial observations that allow for testing the physics of the gravitational interaction. These tests include detection of gravitational waves by advanced LIGO-Virgo antennas, Event Horizon Telescope observations of central relativistic compact objects (RCO) in active galactic nuclei (AGN), X-ray spectroscopic observations of Fe Kα line in AGN, Galactic X-ray sources measurement of masses and radiuses of neutron stars, quark stars, and other RCO. A very important task of observational cosmology is to perform large surveys of galactic distances independent on cosmological redshifts for testing the nature of the Hubble law and peculiar velocities. Forthcoming multimessenger astronomy, while using such facilities as advanced LIGO-Virgo, Event Horizon Telescope (EHT), ALMA, WALLABY, JWST, EUCLID, and THESEUS, can elucidate the relation between Einstein’s geometrical and Feynman’s quantum-field approaches to gravity physics and deliver a new possibilities for unification of gravitation with other fundamental quantum physical interactions.

Relaxing constraints on data acquisition and position detection for trap stiffness calibration in optical tweezers

Optical tweezers find applications in various fields, ranging from biology to physics. One of the fundamental steps necessary to perform quantitative measurements using trapped particles is the calibration of the tweezer's spring constant. This can be done through power spectral density analysis, from forward scattering detection of the particle's position. In this work we propose and experimentally test simplifications to such measurement procedure, aimed at reducing post-processing of recorded data and dealing with acquisition devices that have frequency-dependent electronic noise. In the same line of simplifying the tweezer setup we also present a knife-edge detection scheme that can substitute standard position sensitive detectors.

SciPy 1.0: fundamental algorithms for scientific computing in Python

SciPy is an open-source scientific computing library for the Python programming language. Since its initial release in 2001, SciPy has become a de facto standard for leveraging scientific algorithms in Python, with over 600 unique code contributors, thousands of dependent packages, over 100,000 dependent repositories and millions of downloads per year. In this work, we provide an overview of the capabilities and development practices of SciPy 1.0 and highlight some recent technical developments.

SciPy 1.0: fundamental algorithms for scientific computing in Python

SciPy is an open-source scientific computing library for the Python programming language. Since its initial release in 2001, SciPy has become a de facto standard for leveraging scientific algorithms in Python, with over 600 unique code contributors, thousands of dependent packages, over 100,000 dependent repositories and millions of downloads per year. In this work, we provide an overview of the capabilities and development practices of SciPy 1.0 and highlight some recent technical developments.

SciPy 1.0: fundamental algorithms for scientific computing in Python

SciPy is an open-source scientific computing library for the Python programming language. Since its initial release in 2001, SciPy has become a de facto standard for leveraging scientific algorithms in Python, with over 600 unique code contributors, thousands of dependent packages, over 100,000 dependent repositories and millions of downloads per year. In this work, we provide an overview of the capabilities and development practices of SciPy 1.0 and highlight some recent technical developments.

High-precision laser spot center positioning method for weak light conditions

In the acquisition stage of many space applications, such as the Taiji program, the spot center of weak laser light must be accurately determined. Under weak light conditions, the precision of most traditional positioning methods is greatly affected. In this paper, we present a high-precision laser spot center positioning method based on the theoretical analysis of influence factors of precision. It is shown through experimental study that the method's precision can fulfill the requirement of the Taiji program.

Prospects for observing and localizing gravitational-wave transients with Advanced LIGO, Advanced Virgo and KAGRA

We present our current best estimate of the plausible observing scenarios for the Advanced LIGO, Advanced Virgo and KAGRA gravitational-wave detectors over the next several years, with the intention of providing information to facilitate planning for multi-messenger astronomy with gravitational waves. We estimate the sensitivity of the network to transient gravitational-wave signals for the third (O3), fourth (O4) and fifth observing (O5) runs, including the planned upgrades of the Advanced LIGO and Advanced Virgo detectors. We study the capability of the network to determine the sky location of the source for gravitational-wave signals from the inspiral of binary systems of compact objects, that is binary neutron star, neutron star-black hole, and binary black hole systems. The ability to localize the sources is given as a sky-area probability, luminosity distance, and comoving volume. The median sky localization area (90% credible region) is expected to be a few hundreds of square degrees for all types of binary systems during O3 with the Advanced LIGO and Virgo (HLV) network. The median sky localization area will improve to a few tens of square degrees during O4 with the Advanced LIGO, Virgo, and KAGRA (HLVK) network. During O3, the median localization volume (90% credible region) is expected to be on the order of 10 5 , 10 6 , 10 7 Mpc 3 for binary neutron star, neutron star-black hole, and binary black hole systems, respectively. The localization volume in O4 is expected to be about a factor two smaller than in O3. We predict a detection count of 1 - 1 + 12 ( 10 - 10 + 52 ) for binary neutron star mergers, of 0 - 0 + 19 ( 1 - 1 + 91 ) for neutron star-black hole mergers, and 17 - 11 + 22 ( 79 - 44 + 89 ) for binary black hole mergers in a one-calendar-year observing run of the HLV network during O3 (HLVK network during O4). We evaluate sensitivity and localization expectations for unmodeled signal searches, including the search for intermediate mass black hole binary mergers.

Fundamental Physics Implications for Higher-Curvature Theories from Binary Black Hole Signals in the LIGO-Virgo Catalog GWTC-1

Gravitational-wave astronomy offers not only new vistas into the realm of astrophysics, but it also opens an avenue for probing, for the first time, general relativity in its strong-field, nonlinear, and dynamical regime, where the theory's predictions manifest themselves in their full glory. We present a study of whether the gravitational-wave events detected so far by the LIGO-Virgo scientific collaborations can be used to probe higher-curvature corrections to general relativity. In particular, we focus on two examples: Einstein-dilaton-Gauss-Bonnet and dynamical Chern-Simons gravity. We find that the two events with a low-mass m≈7  M_{⊙} BH (GW151226 and GW170608) place stringent constraints on Einstein-dilaton-Gauss-Bonnet gravity, α_{EDGB}^{1/2}≲5.6  km, whereas dynamical Chern-Simons gravity remains unconstrained by the gravitational-wave observations analyzed.

Reduced-Order Modeling with Artificial Neurons for Gravitational-Wave Inference

Gravitational-wave data analysis is rapidly absorbing techniques from deep learning, with a focus on convolutional networks and related methods that treat noisy time series as images. We pursue an alternative approach, in which waveforms are first represented as weighted sums over reduced bases (reduced-order modeling); we then train artificial neural networks to map gravitational-wave source parameters into basis coefficients. Statistical inference proceeds directly in coefficient space, where it is theoretically straightforward and computationally efficient. The neural networks also provide analytic waveform derivatives, which are useful for gradient-based sampling schemes. We demonstrate fast and accurate coefficient interpolation for the case of a four-dimensional binary-inspiral waveform family and discuss promising applications of our framework in parameter estimation.

Gravitational-wave astronomy: delivering on the promises

Now that LIGO and Virgo have begun to detect gravitational-wave events with regularity, the field of gravitational-wave astronomy is beginning to realize its promise. Binary black holes and, very recently, binary neutron stars have been observed, and we are already learning much from them. The future, with improved sensitivity, more detectors and detectors like LISA in different frequency bands, has even more promise to open a completely hidden side of the Universe to our exploration.This article is part of a discussion meeting issue 'The promises of gravitational-wave astronomy'.

Prospects for observing and localizing gravitational-wave transients with Advanced LIGO, Advanced Virgo and KAGRA

We present possible observing scenarios for the Advanced LIGO, Advanced Virgo and KAGRA gravitational-wave detectors over the next decade, with the intention of providing information to the astronomy community to facilitate planning for multi-messenger astronomy with gravitational waves. We estimate the sensitivity of the network to transient gravitational-wave signals, and study the capability of the network to determine the sky location of the source. We report our findings for gravitational-wave transients, with particular focus on gravitational-wave signals from the inspiral of binary neutron star systems, which are the most promising targets for multi-messenger astronomy. The ability to localize the sources of the detected signals depends on the geographical distribution of the detectors and their relative sensitivity, and [Formula: see text] credible regions can be as large as thousands of square degrees when only two sensitive detectors are operational. Determining the sky position of a significant fraction of detected signals to areas of 5-[Formula: see text] requires at least three detectors of sensitivity within a factor of [Formula: see text] of each other and with a broad frequency bandwidth. When all detectors, including KAGRA and the third LIGO detector in India, reach design sensitivity, a significant fraction of gravitational-wave signals will be localized to a few square degrees by gravitational-wave observations alone.

Distinguishing spin-aligned and isotropic black hole populations with gravitational waves

The direct detection of gravitational waves from merging binary black holes opens up a window into the environments in which binary black holes form. One signature of such environments is the angular distribution of the black hole spins. Binary systems that formed through dynamical interactions between already-compact objects are expected to have isotropic spin orientations (that is, the spins of the black holes are randomly oriented with respect to the orbit of the binary system), whereas those that formed from pairs of stars born together are more likely to have spins that are preferentially aligned with the orbit. The best-measured combination of spin parameters for each of the four likely binary black hole detections GW150914, LVT151012, GW151226 and GW170104 is the 'effective' spin. Here we report that, if the magnitudes of the black hole spins are allowed to extend to high values, the effective spins for these systems indicate a 0.015 odds ratio against an aligned angular distribution compared to an isotropic one. When considering the effect of ten additional detections, this odds ratio decreases to 2.9 × 10^-7 against alignment. The existing preference for either an isotropic spin distribution or low spin magnitudes for the observed systems will be confirmed (or overturned) confidently in the near future.

Impact of Bayesian Priors on the Characterization of Binary Black Hole Coalescences

In a regime where data are only mildly informative, prior choices can play a significant role in Bayesian statistical inference, potentially affecting the inferred physics. We show this is indeed the case for some of the parameters inferred from current gravitational-wave measurements of binary black hole coalescences. We reanalyze the first detections performed by the twin LIGO interferometers using alternative (and astrophysically motivated) prior assumptions. We find different prior distributions can introduce deviations in the resulting posteriors that impact the physical interpretation of these systems. For instance, (i) limits on the 90% credible interval on the effective black hole spin χ_{eff} are subject to variations of ∼10% if a prior with black hole spins mostly aligned to the binary's angular momentum is considered instead of the standard choice of isotropic spin directions, and (ii) under priors motivated by the initial stellar mass function, we infer tighter constraints on the black hole masses, and in particular, we find no support for any of the inferred masses within the putative mass gap M≲5  M_{⊙}.

Maximum Redshift of Gravitational Wave Merger Events

Future generations of gravitational wave detectors will have the sensitivity to detect gravitational wave events at redshifts far beyond any detectable electromagnetic sources. We show that if the observed event rate is greater than one event per year at redshifts z≥40, then the probability distribution of primordial density fluctuations must be significantly non-Gaussian or the events originate from primordial black holes. The nature of the excess events can be determined from the redshift distribution of the merger rate.

GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral

On August 17, 2017 at 12∶41:04 UTC the Advanced LIGO and Advanced Virgo gravitational-wave detectors made their first observation of a binary neutron star inspiral. The signal, GW170817, was detected with a combined signal-to-noise ratio of 32.4 and a false-alarm-rate estimate of less than one per 8.0×10^{4}  years. We infer the component masses of the binary to be between 0.86 and 2.26  M_{⊙}, in agreement with masses of known neutron stars. Restricting the component spins to the range inferred in binary neutron stars, we find the component masses to be in the range 1.17-1.60  M_{⊙}, with the total mass of the system 2.74_{-0.01}^{+0.04}M_{⊙}. The source was localized within a sky region of 28  deg^{2} (90% probability) and had a luminosity distance of 40_{-14}^{+8}  Mpc, the closest and most precisely localized gravitational-wave signal yet. The association with the γ-ray burst GRB 170817A, detected by Fermi-GBM 1.7 s after the coalescence, corroborates the hypothesis of a neutron star merger and provides the first direct evidence of a link between these mergers and short γ-ray bursts. Subsequent identification of transient counterparts across the electromagnetic spectrum in the same location further supports the interpretation of this event as a neutron star merger. This unprecedented joint gravitational and electromagnetic observation provides insight into astrophysics, dense matter, gravitation, and cosmology.

Bounding the Speed of Gravity with Gravitational Wave Observations

The time delay between gravitational wave signals arriving at widely separated detectors can be used to place upper and lower bounds on the speed of gravitational wave propagation. Using a Bayesian approach that combines the first three gravitational wave detections reported by the LIGO Scientific and Virgo Collaborations we constrain the gravitational waves propagation speed c_{gw} to the 90% credible interval 0.55c<c_{gw}<1.42c, where c is the speed of light in vacuum. These bounds will improve as more detections are made and as more detectors join the worldwide network. Of order 20 detections by the two LIGO detectors will constrain the speed of gravity to within 20% of the speed of light, while just five detections by the LIGO-Virgo-Kagra network will constrain the speed of gravity to within 1% of the speed of light.

GW170104: Observation of a 50-Solar-Mass Binary Black Hole Coalescence at Redshift 0.2

We describe the observation of GW170104, a gravitational-wave signal produced by the coalescence of a pair of stellar-mass black holes. The signal was measured on January 4, 2017 at 10∶11:58.6 UTC by the twin advanced detectors of the Laser Interferometer Gravitational-Wave Observatory during their second observing run, with a network signal-to-noise ratio of 13 and a false alarm rate less than 1 in 70 000 years. The inferred component black hole masses are 31.2_{-6.0}^{+8.4}M_{⊙} and 19.4_{-5.9}^{+5.3}M_{⊙} (at the 90% credible level). The black hole spins are best constrained through measurement of the effective inspiral spin parameter, a mass-weighted combination of the spin components perpendicular to the orbital plane, χ_{eff}=-0.12_{-0.30}^{+0.21}. This result implies that spin configurations with both component spins positively aligned with the orbital angular momentum are disfavored. The source luminosity distance is 880_{-390}^{+450}  Mpc corresponding to a redshift of z=0.18_{-0.07}^{+0.08}. We constrain the magnitude of modifications to the gravitational-wave dispersion relation and perform null tests of general relativity. Assuming that gravitons are dispersed in vacuum like massive particles, we bound the graviton mass to m_{g}≤7.7×10^{-23}  eV/c^{2}. In all cases, we find that GW170104 is consistent with general relativity.