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The Pennsylvania State University The Graduate School

DETECTING GRAVITATIONAL WAVES FOR

MULTI-MESSENGER ASTRONOMY

A Dissertation in Physics by Cody Messick

Submitted in Partial Fulﬁllment of the Requirements for the Degree of

Doctor of Philosophy

August 2019 The dissertation of Cody Messick was reviewed and approved∗ by the following:

Chad Hanna Associate Professor of Physics & Astronomy and Astrophysics Dissertation Advisor, Chair of Committee

Doug Cowen Professor of Physics & Astronomy and Astrophysics

Irina Mocioiu Associate Professor of Physics

Derek Fox Associate Professor of Astronomy and Astrophysics

Richard Robinett Professor of Physics Associate Head for Undergraduate and Graduate Studies

∗Signatures are on ﬁle in the Graduate School.

ii Abstract

Gravitational waves, ripples in space-time that cause the physical distance between points to change in time, were first predicted by Albert Einstein in the early twentieth century. One of the most promising sources of gravitational waves was thought to be the merger of binary neutron stars, which were also believed to generate extremely energetic bursts of light known as gamma-ray bursts and an optical transient known as a kilonova. Large interferometers capable of measuring distances of 10−18 m were built to detect gravitational waves. My dissertation research has been focused on detecting gravitational waves rapidly for the purpose of searching for electromagnetic or other signals from merging neutron star binaries. I contributed to the first-ever detection of gravitational waves in 2015, and every detection since. My work to rapidly identify new gravitational wave detections led to the first joint multi-messenger detection made in 30 years on August 17, 2017, when gravitational waves from a binary neutron star merger were observed in coincidence with a short gamma-ray burst, which led astronomers around the world to point their instruments at the region of the sky where the signal was estimated to have came from in time to observe counterparts across several electromagnetic bands. In this dissertation I will discuss each of these historical detections and the analysis I co-developed and helped perform.

iii Table of Contents

List of Figures viii

List of Tables xvii

Acknowledgments xxv

Chapter 1 Introduction to Gravitational-Wave Astronomy 1 1.1 Einstein’s General Theory of Relativity ...... 3 1.1.1 Deﬁnitions ...... 3 1.1.2 Curvature ...... 7 1.1.3 Einstein’s Field Equations ...... 8 1.1.3.1 The Newtonian Limit ...... 9 1.2 Gravitational Waves ...... 11 1.2.1 Quadrupole Formula ...... 11 1.2.1.1 Quadrupole Formula of Binary ...... 15 1.3 Compact Binaries ...... 17 1.3.1 Merger Mechanism ...... 17 1.3.2 Merger Rates ...... 20 1.3.3 Waveform Approximations ...... 23 1.4 The LIGO Detectors ...... 24 1.5 Introduction to Compact Binary Data Analysis Methods ...... 27 1.5.1 Matched Filtering ...... 29 1.5.1.1 Standard Normal Noise Distribution ...... 29 1.5.1.2 Normal Noise Distribution ...... 30 1.5.1.3 Real LIGO Data ...... 35 1.5.2 The p-value ...... 36 1.5.3 Template Banks ...... 39 1.5.4 Horizon Distance, Eﬀective Distance, and Range ...... 40 1.6 The GstLAL-based Inspiral Pipeline ...... 41

iv 1.6.1 The LLOID Algorithm ...... 41 1.6.1.1 Critical Sampling ...... 42 1.6.1.2 Singular Value Decomposition ...... 43 1.6.1.3 The LLOID Template Decomposition ...... 44 1.6.2 Signal-Consistency Test Parameter ξ2 ...... 44 1.6.3 Trigger Generation ...... 46 1.6.4 GstLAL’s Likelihood-Ratio ...... 47 1.6.5 Estimating Signiﬁcance ...... 49

Chapter 2 GW150914 52 2.1 Introduction ...... 52 2.2 Methods ...... 53 2.2.1 Template Bank ...... 53 2.2.2 Likelihood-ratio ...... 54 2.3 Initial CBC Results ...... 54 2.4 Published CBC Results ...... 55 2.4.1 p-value sanity check ...... 57 2.5 Comparison to Parameter Estimation Results ...... 58 2.6 Estimated Rates ...... 59

Chapter 3 GW151226 65 3.1 Introduction ...... 65 3.2 Methods ...... 65 3.3 Low-Latency Detection ...... 66 3.3.1 Multi-Messenger Follow-Up ...... 66 3.4 Published Oﬄine Results ...... 66 3.5 Comparison to Parameter Estimation Results ...... 68 3.6 Estimated Rates ...... 69

Chapter 4 GW170104 72 4.1 Introduction ...... 72 4.2 Methods ...... 72 4.2.1 Template Bank ...... 72 4.2.2 Likelihood-ratio ...... 74 4.3 Published Oﬄine Results ...... 76 4.4 Comparison to Parameter Estimation Results ...... 78 4.5 Estimated Rates ...... 80

v Chapter 5 GW170814 83 5.1 Introduction ...... 83 5.2 Methods ...... 84 5.2.1 Template Bank ...... 84 5.2.1.1 Low-latency Bank ...... 84 5.2.1.2 Oﬄine Bank ...... 84 5.2.2 Likelihood-ratio ...... 85 5.3 Low-Latency Detection ...... 86 5.3.1 Multi-Messenger Follow-Up ...... 87 5.4 Published Oﬄine Results ...... 88 5.5 Comparison to Parameter Estimation Results ...... 89 5.6 Estimated Rates ...... 90

Chapter 6 GW170817 93 6.1 Introduction ...... 93 6.2 Methods ...... 94 6.2.1 Likelihood-ratio ...... 94 6.3 Low-Latency Detection ...... 95 6.3.1 Multi-Messenger Follow-Up ...... 96 6.4 Published Oﬄine Results ...... 98 6.5 Comparison to Parameter Estimation Results ...... 99 6.6 Estimated Rates ...... 100

Chapter 7 GWTC-1 109 7.1 Introduction ...... 109 7.2 Methods ...... 109 7.2.1 Likelihood-ratio ...... 110 7.3 Published Results ...... 112 7.4 Comparison to Parameter Estimation Results ...... 116 7.5 Estimated Rates ...... 118

Chapter 8 Low-Latency Alerts in O3 124 8.1 Introduction ...... 124 8.2 Methods ...... 124 8.2.1 Bank ...... 124 8.2.2 Likelihood-ratio ...... 125

vi 8.2.3 Itacac ...... 125 8.3 Public Alerts To Date ...... 127 8.4 Current State of the Field ...... 129 8.5 Conclusion ...... 130

Bibliography 131

vii List of Figures

1.1 A schematic of the basic interferometer design of ground-based interferometric detectors. Naively, the detector can be thought of as a Michelson interferometer, though a naive Michelson interferometer would not be sensitive enough to detect gravitational waves. This ﬁgure originally appeared in Ref. [1]...... 25

1.2 An example of the whitening process. The plot on the left shows 8 seconds of stationary, Gaussian noise that is not white. The plot on the right shows the same data after it has been whitened. The data used to create this plot were downloaded from the Gravitational Wave Open Science Center [2]...... 33

1.3 A noise transient, commonly referred to as a glitch, from December 22, 2015. Whitened data should have a mean of zero and a variance of one, thus the peak amplitude of this glitch is a several-hundred σ excursion from stationary, Gaussian data. Whitened data should also be uncorrelated from one time sample to the next, which is clearly not true near the glitch. Glitches such as this occur several times an hour in real LIGO data. The data used to create this plot were downloaded from the Gravitational Wave Open Science Center [2]...... 36

viii 1.4 The Power Spectral Density (PSD) measured from data taken from the morning of December 22, 2015, at the Hanford, WA detector. The PSD represents the average square-magnitude of the noise as a function of frequency, and is used to weight the matched-ﬁlter such that frequency bins where the noise is lower are more heavily weighted than bins where the noise is higher. The data used to create this plot were downloaded from the Gravitational Wave Open Science Center [2]...... 37

1.5 A plot of the component masses of a template bank. Every point represents a diﬀerent template used for matched-ﬁltering. More details can be found on this template bank in Sec. 4.2.1. This plot was originally published in Ref. [3]...... 40

1.6 The template decomposition used by the Low-Latency Online In- spiral Detection (LLOID) algorithm. The top of the figure shows three inspiral templates, with sections color-coded to represent the time-slices, which are sampled at different rates. The bottom of the figure shows examples of the orthogonal basis vectors returned by the singular value decomposition (SVD) of the sliced templates. The templates in the SVD basis are convolved with the data in the matched-filtering process, though the final result of the LLOID algorithm is the Signal-to-Noise Ratio (SNR) time-series in the original basis, sampled at the sample rate of the data. This figure originally appeared in Ref. [4]...... 45

1.7 Simulated likelihood-ratios before and after clustering. Clustering removes candidates such that only the highest likelihood-ratio in any given 8 second window is preserved. This does not significantly affect the distribution of large likelihood-ratios, but it causes an extinction effect where too many low likelihood-ratio candidates are removed to reproduce the low likelihood-ratio regime without additional modeling...... 50

ix 2.1 Plots of the number of candidates with inverse false-alarm rates (IFAR) above the threshold on the x-axis. The top plot is known as a closed box plot, and is from an analysis where the L1 data has been shifted in time so that any coincidences that form between the two detectors are most likely noise. The fact that the observed distribution, the solid line, follows the expected distribution tells us that the false alarm rates being assigned are accurate for noise (i.e. this is veriﬁcation that a false alarm rate of e.g. 1 per day actually means one only expects something at least this signiﬁcant to happen once a day). The bottom plot shows the open box analysis, showing the real results. GW150914 falls far outside the expected region for noise with an IFAR that is 26 orders of magnitude greater than the right edge of the plot. The straight line on the left is due to the analysis only considering candidates above some log L threshold, any events with lower ranking-statistic values are simply assigned the FAR associated with the lowest ranking-statistic value considered. 62

2.2 The IFAR plots for the analysis that the published GW150914 results are from. The top plot is produced by shifting the L1 timestamps and then analyzing the coincidences normally, providing a check that the FAR estimation is accurate. The fact that the time shifted candidates lie within the expected region means the FARs are accurate for noise. The bottom plot shows the open box results, where data has not been time shifted. There are two candidates that stand out, the marginal candidate GW151012 (which was not granted a GW title until 2018 [5]) and GW150914. The IFAR of GW150914 is 26 orders of magnitude greater than the right edge of the plot...... 63

x 2.3 The (ρ, ξ2/ρ2) PDF under the noise hypothesis, marginalized across the entire template bank. The PDFs shown in the warm color maps use the GstLAL-based inspiral pipeline’s default noise model, which is estimated from non-coincident triggers from each detector. The regions show in the blue color map are from the PDF estimated using a noise model that includes observed candidates; only regions that are signiﬁcantly diﬀerent from the default noise model PDF are shown. The only regions shown in blue are from GW150914. Part of the procedure to estimate the PDF involves smoothing the histograms, hence the counts from GW150914 are spread out into a few bins and produce the large “protrusion” at the bottom. This is a plot I made, originally for Ref. [6]...... 64

2.4 The skymap estimated rapidly using BAYESTAR [7] and the SNRs found by the GstLAL-based inspiral pipeline (left) compared to the ﬁnal skymap estimated from the parameter estimation runs (right). The overlap region is 360 deg2...... 64

3.1 The expected counts above a given inverse FAR (IFAR) threshold for the analysis the published GW151226 significance estimate was taken from. This analysis was over all of the coincident data taken during O1, using the template bank described in Table 2.1. The closed box plot (top) is generated the same way as the previous closed box plots (see e.g. Fig. 2.2) and shows that the FAR calculation produces accurate FARs for noise. The open box plot (bottom) cuts off at too low of an IFAR to see GW151226 or GW150914. The third loudest candidate in O1 was GW151012, then named LVT151012 to reflect its marginal significance...... 70

3.2 The original skymap estimated by BAYESTAR [7] using the low- latency detection of GW151226 (left), and the ﬁnal skymap from a full Bayesian analysis (right)...... 71

4.1 A plot of the component masses of the template bank used by the analysis that produced the published GW170104 signiﬁcance estimate, originally published in Ref. [3]...... 73

xi 4.2 ∆t distributions under the signal hypothesis, estimated from the analysis of injected binary black hole mergers used to estimate merger rates at the end of O1 in Ref. [8]. The unsmoothed data is a histogram, the smoothed data is then the histogram smoothed with a Gaussian kernel, and the ﬁt is computed using Chebyshev polynomials...... 77

4.3 A sanity check of the numerical method used to compute the ∆t signal-model term in the likelihood-ratio. The line labeled “original fit” is a Chebyshev polynomial computed using coefficients that were fit to the observed data. The predicted fit line is the result of using Chebyshev polynomials to estimate the Chebyshev coefficients used in the original fit as a function of ρ...... 78

4.4 The ∆t and ∆φ signal models used in the likelihood-ratio numerator, computed using models whose coeﬃcients have been ﬁt to the distributions observed in the population of injections used to estimate binary black hole merger rates in O1. Figure from Ref. [9]. 79

4.5 The counts above a given IFAR threshold vs the IFAR threshold. The closed box results (top) were generated using the method described in Fig. 2.2 and should be interpreted in the same way, i.e. we know the analysis is assigning accurate FARs to noise candidates because the observed closed box results lie within the expected region. The open box (bottom) show two candidates that appear to stick out above noise, GW170104, which has an IFAR 3 orders of magnitude higher than the right edge of the plot , and a 1.4σ candidate that disappeared in subsequent analyses...... 81

4.6 The skymap estimated by BAYESTAR [7] using SNRs from the GstLAL-based inspiral pipeline is on the left, and the skymap estimated from the ﬁnal parameter estimation run which performs a full Bayesian analysis is on the right. The data used to generate the BAYESTAR skymap has never been released publicly, and in fact did not come from an analysis over the full template bank; the data are the result of a sanity check that was done where templates near the expected GW170104 parameters were matched-ﬁltered and the output was ranked using the state of the low-latency background at the time of the analysis (roughly a day after GW170104). . . . . 82

xii 5.1 The component masses of the template bank used in the low-latency analysis that detected GW170814 and GW170817 (Ch. 6). The template bank is identical to the template bank described in Ta- ble 4.1 except that it has a total mass upper cutoﬀ of 150 M . This ﬁgure originally appeared in Ref. [3]...... 85

5.2 The template bank used by the analysis that computed the published GW170814 significance. It is a modified version of the bank described in Table 4.1. The high mass region now has a minimum match of 0.98, up from 0.97. 1000 additional templates were then added to the high mass region in a grid that is uniformly spaced in component mass. This figure originally appeared in Ref. [3]. . . . . 86

5.3 The number of candidates above an IFAR threshold plotted against the IFAR threshold. The closed box results (top) are generated and should be interpreted the same way as the closed box results described in Fig. 2.2. Once again, the fact that the observed closed box counts lie within the expected region is evidence the analysis is assigning accurate FARs. There are two signals in the open box plot (bottom), both to the right of the visible region: GW170814, which has an IFAR 13 orders of magnitude larger than the right edge of the plot, and GW170817, which has an IFAR 21 orders of magnitude larger than the right edge of the plot...... 91

5.4 The initial skymap is on the left, which was estimated rapidly [7] using the low-latency detection. A bug was later identified which resulted in a systematic shift of the skymap by a small amount, though it has since been fixed. The final skymap is on the right, estimated by a full Bayesian analysis...... 92

xiii 6.1 A comparison of the original (black line) and updated (dashed red line) horizon distance calculations. The original method computed the horizon distance using the PSD at the time of the candidate, which meant that when a detector characterization data-quality ﬂag vetoed the glitch in L1 just before merger of GW170817, the horizon distance for L1 was set to zero and the low-latency software pipeline treated L1 as oﬀ. The updated algorithm computes a volume-weighted average over the previous 5 minutes, so that short vetoes will not prevent detections. This new algorithm is robust to occasional bursts of noise, and instead returns a lower sensitivity during periods when the glitch rate is high, which is a more desirable behavior...... 105

6.2 The time-frequency plots automatically generated when GW170817 was uploaded to GraceDB [10]. A faint track is visible in H1, and a much louder track is visible in L1 behind a “glitch,” a common name for transient noise. The glitch is the reason GW170817 was identiﬁed as a single-detector candidate...... 106

6.3 The original skymap estimated by BAYESTAR [7] (left), and the 3 detector skymap estimated by BAYESTAR (right). The original skymap simply shows the antenna pattern of H1 at the time of the detection, while the 3-detector skymap had a 90% conﬁdence area of 31 deg2...... 106

6.4 Whitened strain data from L1 at the time of the glitch that occurred before the merger of GW170817. The “Veto Gate” refers to the GStreamer element removes vetoed times, the “Autogate” refers to the algorithm that self-vetoes times where whitened strain data passes a set threshold [4]. The full second of data at 12:41:03 UTC was removed instead of only removing data with 0.03125 s (plus some padding) of 12:41:03.40625 UTC. This was ﬁxed in subsequent analyses, for now the only result is a decreased L1 SNR and a worse ξ2...... 107

6.5 The posterior component mass distributions for GW170817, for both the low-spin prior (motivated by the observed binary neutron star population) and the high-spin prior. Both priors show good support for equal component masses. This ﬁgure originally appeared in Ref. [11]...... 108

xiv 6.6 The average space-time, hVTi, volume the detectors are sensitive to is estimated by injecting simulated signals and recording how the time-averaged fraction that are recovered as a function of distance. The plot on the left shows the estimates using the original injection analysis algorithm. The plot on the right shows the estimate obtained when only looking at speciﬁc regions of the template bank for injections. These regions need to be large enough to include the majority of templates expected to ring up from an injection; I developed an algorithm that chose what region of the template bank to use when searching for analyses based on the parameters of the injections that estimates an average space-time volume accurate to within 1% of the calculation that uses the entire template bank while running in less time and using less computational resources. . 108

7.1 The number of candidates above a given log-likelihood-ratio threshold vs that threshold. We are now in the regime where the signal population is loud enough that the observed cumulative counts follow a combined noise-and-signal population, thus the signal model and the noise-plus-signal are plotted in addition to the noise model, to check if the observed distribution is as expected. The right plot is zoomed in at the left edge of the left plot, to show that the closed box results (obtained and interpreted the same way as in Fig. 2.2) are in agreement with the noise model. The analysis has become sensitive enough that it can now detect loud signals even in the closed box, pairing the loud signal with a random peak. As a result, the top two loudest closed box candidates were associated with GW170817 and GW150914, respectively, and had to be manually removed to produce a signal-free population to check the noise-model. The shaded regions can be thought of as the “1ish” and “2ish” σ regions. The ﬁgure on the left is a ﬁgure I created for Ref. [5]...... 114

xv 8.1 The analytical signal model being used in low-latency alerts in O3 at the time of writing. The binary neutron star region is assumed to follow a Gaussian distribution in chirp mass, with µ = 1.2 M and σ = .25 M , while the binary black hole region is assumed to follow a power-law distribution in chirp mass with an index of −1.5. The relative FARs in these plots are only to show how their FAR would be aﬀected relative to other templates in the plot. The left plot is zoomed in on low masses, while the right plot shows the entire template bank. The plots were both normalized individually, thus the normalizations of the color bars are not expected to be the same...... 126

8.2 A high-level depiction of the workﬂow in the GstLAL-based inspiral pipeline. Before O3, SNR time-series were converted to triggers one detector at a time by an in-house GStreamer element named itac, meaning that sub-threshold SNR information is lost before uploading to GraceDB. Now the SNR time-series from all detectors are fed into a new in-house GStreamer element named itacac, which converts the time-series from diﬀerent detectors to triggers at the same time. This means that if one (or more) detectors have an SNR below the trigger threshold, SNR from that detector can still be saved and uploaded to GraceDB [10] to be used in rapid localization. 127

xvi List of Tables

2.1 The parameters of the template bank used in the analysis that estimated the signiﬁcance of GW150914...... 54

2.2 Information about GW150914 from the initial oﬄine GstLAL-based inspiral pipeline analysis, which analyzed 5 days of coincident data for compact binaries in the range listed in Table 2.1...... 55

2.3 Results from the analysis that the published significance estimate of GW150914 came from [6]. The analysis considered roughly 16 days of coincident data, and looked for signals from compact binary’s whose properties lay in the template bank described in Table 2.1. The first candidate is Gw150914, the second is what is now known as GW151012. At the time, it was not significant enough to warrant a detection claim. The values listed in this table differ from what was published because the GstLAL-based inspiral pipeline’s default noise model is used here, which estimates the distribution of log L in noise using non-coincident triggers. Due no other data producing triggers similar to GW150914 in either detector, the pipeline assigned it an extremely low p-value. The collaboration requested that the noise model also include observed candidates, and published those numbers. GW150914 is so different from the noise of the detectors however that that significance was still greater than 5σ even when it was in its own background. This dissertation will only contain significance estimates from the default noise model, as it minimizes signal contamination...... 56

xvii 2.4 The point estimates of parameters from the template GstLAL-based inspiral pipeline chose as the most signiﬁcant, compared to the results of a full Bayesian analysis that explored a much denser and higher-dimension parameter space but only considers data near the time of the candidate. ai is the dimensionless spin magnitude, which can vary from 0 to 1. The parameter estimation results are in the detector frame, which has been redshifted, making the masses appear larger than they are in the source frame by a factor of (1 + z). The template parameters do not fall inside the 90% conﬁdence intervals for any of the masses, and only one of the spins. It is unclear why, though there are possible reasons (e.g. the signal does not contain many cycles to help distinguish it from other waveforms, this may conspire with the fact that the search pipelines only consider a sparse, low-dimensional parameter space to distort the parameter estimates, or to go in the other direction, perhaps the assumption of Gaussian noise by the parameter estimation analyses does not accurately describe the noise as well as the GstLAL-based inspiral pipeline’s noise model; there is no evidence for either hypothesis). . 59

3.1 The significance and template parameters GW151226 was identified with in low-latency. The analysis began analyzing low-latency data from the detectors on November 15, 2015, and used the template bank described in Table 2.1. GW151226 was identified and uploaded to GraceDB [10] within 70 seconds of passing through the detector...... 66

3.2 The significance estimate and parameters of the best-fit template for GW151226, taken from the analysis that produced the published significance estimate [12]. As described below Table 2.3, the noise model used in this dissertation is the GstLAL-based inspiral pipeline’s default noise model that minimizes signal contamination, which is different than the noise model used in the paper...... 67

xviii 3.3 The parameters of the template the GstLAL-based inspiral pipeline used to identify GW151226 compared to the detector-frame median and 90% conﬁdence intervals estimated by parameter estimation pipelines. All of the point estimates fall within the 90% conﬁdence intervals. GW151226 was in the detector’s sensitive frequency band for longer than GW150914, which may explain the improvement in parameter estimation...... 69

4.1 The parameters of the template bank used by the analysis that produced the published GW170104 signiﬁcance estimate...... 73

4.2 The significance estimates and parameters of the two most significant candidates found by the analysis that produced the published GW170104 significance estimate [13]. The loudest candidate is GW170104, the second loudest occurred at 05:58:49 UTC on Jan- uary 17, 2017, but subsequent analyses substantially decreased its significance and thus it is believed to be noise. As explained in Ta- ble 2.3, the noise model used to estimate the significance estimates listed here is the default used by the GstLAL-based inspiral pipeline which minimizes signal contamination, while the noise model used to estimate the published results used a noise model that added observed candidates to the noise...... 78

4.3 A comparison of the parameters from the template the GstLAL- based inspiral pipeline associated with GW170104 and the median and 90% credible region detector-frame parameter estimates from the parameter estimation pipelines, which implement a full Bayesian analysis. The chirp mass and the spin magnitude estimates are within the 90% conﬁdence regions, though the component masses and total mass are not. To zeroth order though, the behavior of the waveform is proportional to the chirp mass, thus if only one parameter is estimated correctly it is expected to be the chirp mass...... 80

xix 5.1 The significance and template parameters of the most significant low-latency candidate associated with GW170814. At the time of detection, Virgo triggers were generated to check for coincidence but were not incorporated into the likelihood-ratio, i.e. the Virgo trigger did not affect the significance estimate. It did, however, improve the localization [14]. The noise model used to estimate the significance estimated here is again the default noise model that minimizes signal contamination as opposed to the noise model used to estimate the published significance estimates which includes observed candidates, as described in Table 2.3...... 87

5.2 The significance and template parameters of the most significant candidate associated with GW170814, as determined by an analysis of roughly 5 days of coincident LIGO data. Virgo data were not used in the significance estimate, though they were used for rapid localization [7] and parameter estimation. As Virgo data are not used to estimate significance, they were not included in the offline analysis, and hence there is no Virgo SNR or ξ2. The noise model used to estimate the significance estimated here is the same as was used previously, which differs from the noise model used to estimate the published significance, see Table 2.3...... 89

5.3 The parameters of the template associated with the most significant GW170814 candidate as reported by the GstLAL-based inspiral pipeline compared to the detector-frame median and 90% credible intervals found by the parameter estimation results. The chirp mass is the only template parameter that falls within the 90% credible region found by the parameter estimation analysis, though the leading order behavior of the inspiral waveform scales with chirp mass so that is not surprising...... 89

6.1 The signiﬁcance estimate and template parameters associated with the most signiﬁcant low-latency GW170817 candidate, which was found as a single-detector H1 candidate...... 95

xx 6.2 The significance and template parameters of the most significant candidate associated with GW170817. As described in Table 2.3, the noise model used to estimate the significance here is different than the noise model used to produce the published significance, but this noise model is the default for the GstLAL-based inspiral pipeline and minimizes signal contamination...... 98

6.3 Comparison of the template parameters associated with the most significant GW170817 candidate identified by the GstLAL-based inspiral pipeline and the detector-frame parameters determined by a full Bayesian analysis [11]. The chirp mass is listed to more significant digits than in previous detections because GW170817 has far more cycles than any of the binary black hole merger signals, allowing us to constrain the chirp mass much more than in previous detections. The component mass ranges are the 90% confidence region that includes the equal mass case (see Fig. 6.5), due to the posterior distributions showing substantial support for an equal mass system. There is a large degeneracy between the mass ratio, q = m1/m2, and the effective spin parameter, χ, so two different spin priors were used: a low-spin prior motivated by spins of observed binary neutron stars, and a high-spin prior loosely motivated by physical models and set by technical limitations. The point estimates from the GstLAL-based inspiral pipeline fall within the 90% credible region of the high-spin priors, though the chirp mass estimate agrees with the range estimated using both spin priors. 100

7.1 The significance and template parameters associated with all candidates with FARs estimated by the GstLAL-based inspiral pipeline to be below 1 per 30 days. The Poisson probability of observing 14 candidates from noise is 2.7σ, thus it is possible all of the marginal candidates are noise, however it is also possible one or more of them are marginal signals. As described in Table 2.3, the noise model used to produce the significance estimates is different than the noise model used to produce the published estimates, though this noise model has lower signal contamination...... 115

xxi 7.2 The signiﬁcance and template parameters associated with the most signiﬁcant GW150914 candidate found in the catalog analysis compared to the updated detector-frame median and 90% credible intervals from parameter estimation. As before (Table 2.4), most of the parameter point estimates are not within the 90% credible interval. Posterior samples hosted by the LIGO Open Science Center [2] were used to compute numbers shown here...... 117

7.3 The signiﬁcance and template parameters associated with the most signiﬁcant GW151012 candidate found in the catalog analysis compared to the updated detector-frame median and 90% credible intervals from parameter estimation. All of the point estimates are within the 90% credible intervals. Posterior samples hosted by the LIGO Open Science Center [2] were used to compute numbers shown here...... 117

7.4 The signiﬁcance and template parameters associated with the most signiﬁcant GW151226 candidate found in the catalog analysis compared to the updated detector-frame median and 90% credible intervals from parameter estimation. As before (Table 3.3), all of the point estimates are within the 90% credible intervals. Posterior samples hosted by the LIGO Open Science Center [2] were used to compute numbers shown here...... 118

7.5 The signiﬁcance and template parameters associated with the most signiﬁcant GW170104 candidate found in the catalog analysis compared to the updated detector-frame median and 90% credible intervals from parameter estimation. As before (Table 4.3), the chirp mass and spin point estimates are within the 90% credible intervals, while the total mass and component mass estimates are not. Pos- terior samples hosted by the LIGO Open Science Center [2] were used to compute numbers shown here...... 118

xxii 7.6 The significance and template parameters associated with the most significant GW170729 candidate found in the catalog analysis compared to the detector-frame median and 90% credible intervals from parameter estimation. All of the parameter point estimates are within the 90% credible intervals, although GW170729 is the qui- etest signal observed (Table 7.1), and as such has broader 90% confidence intervals than the other detections. Posterior samples hosted by the LIGO Open Science Center [2] were used to compute numbers shown here...... 119

7.7 The signiﬁcance and template parameters associated with the most signiﬁcant GW170809 candidate found in the catalog analysis compared to the detector-frame median and 90% credible intervals from parameter estimation. All of the parameter point estimates are within the 90% credible intervals. Posterior samples hosted by the LIGO Open Science Center [2] were used to compute numbers shown here...... 119

7.8 The signiﬁcance and template parameters associated with the most signiﬁcant GW170814 candidate found in the catalog analysis compared to the updated detector-frame median and 90% credible intervals from parameter estimation. Unlike before (Table 5.3), none of the point estimates are within the 90% credible interval. There is no obvious reason this would happen, though it is possible there was a candidate with closer parameters that was slightly suppressed by noise in the Virgo detector (where the signal was barely loud enough to pass our SNR threshold), which is the expected reason GW170814 was recovered as a double-detector candidate instead of a triple-detector candidate. Posterior samples hosted by the LIGO Open Science Center [2] were used to compute numbers shown here. 120

7.9 The significance and template parameters associated with the most significant GW170818 candidate found in the catalog analysis compared to the detector-frame median and 90% credible intervals from parameter estimation. The chirp mass, total mass, and effective spin point estimates fall within the 90% credible regions. Posterior samples hosted by the LIGO Open Science Center [2] were used to compute numbers shown here...... 120

xxiii 7.10 The signiﬁcance and template parameters associated with the most signiﬁcant GW170823 candidate found in the catalog analysis compared to the detector-frame median and 90% credible intervals from parameter estimation. Only the lighter component mass falls within the 90% credible region, though there is no obvious reason why. Posterior samples hosted by the LIGO Open Science Center [2] were used to compute numbers shown here...... 121

7.11 The significance and template parameters associated with the most significant GW170817 candidate found in the catalog analysis compared to the detector-frame 90% credible intervals for the component masses and the median and 90% credible intervals for the other values from parameter estimation. As before (Table 6.3), the parameter estimation numbers were computed using two different spin priors: a low-spin prior motivated by the observed population of binary neutron star spins, and a high-spin prior. All of the parameter point estimates are within the high-spin prior 90% credible intervals. Posterior samples hosted by the LIGO Open Science Center [2] were used to compute numbers shown here...... 121

xxiv Acknowledgments

First and foremost, I have to acknowledge all of my parents’ hard work. If not for their guidance, I would not be where I am today. My parents were not alone in this however, they had help from my both my maternal and paternal grandparents, Bill and Patricia Messick and Ann and Jerry Pongon. I would not be the man I am if not for the amazing support system I grew up with. While my family is the reason I discovered a passion for learning, it was my freshman physics professor, Richard Shamrell, that helped me discover my passion for physics. That passion was challenged when I began taking physics classes at the University of Washington, which were more difficult then anything I’d ever attempted. Thankfully, I once again found an amazing support system, this time in the form of a physics study group: Ryan Beaty, William Johnson, Patrick Godwin, Jeff Schueler, Tim Large, Taryn Black, Meagan Albright, Cliff Plesha, Loc Hua, Brenda Bushell, and Austin Piehl. I will forever cherish my memories of studying on the fourth floor of the physics building and oscillating between cursing our ridiculous homework and excitedly discussing whatever new physics we’d learned that week. Graduate school was another beast entirely. I met amazing people to room with when I first arrived, Miguel Fernandez and James DeLaunay, which made the first year bearable. Stephen Walczyk, Miguel, and James made doing quantum homework the night before it was due possible, and sometimes even enjoyable. Miguel and Ryan Everett made learning the LIGO ropes the most fun I’ve ever had in on-the-job training. Colin Turley, Alan Coleman, Miguel, and James were my board game group and my closest academic group of friends. After classes end in graduate school, most people drift apart, but Kelly Malone is one of the few people I didn’t work or game with that I needed to stay in touch with, mainly to bicker. The graduate school experience is defined by the student’s advisor. I consider myself extremely fortunate to have been Chad Hanna’s student. I owe a lot to Chad for guiding me through the collaboration and teaching me how to be a scientist.

xxv Chad also has impeccable taste in employees. Duncan Meacher, Ryan Everett, Ryan Magee, Sydney Chamberlin, Alex Pace, and Patrick Godwin made research fun instead of a job. While I loved my work environment, nobody can work all the time, so I needed people like Shane Noey and Robert Etchells in my life to help me decompress and to remind me that there’s more to life than gravitational waves. These acknowledgements began, and will end, with family. I want to begin with my mother-in-law, Carrie Cannon, who has shown nothing but support for my decision to take her daughter across the country. She would even send me home with care packages, usually her lasagna made with an old family recipe, my favorite meal. My uncle, Scott Wildman, changed my world when he introduced me to the Grateful Dead. Briana Mahoney, one of my best friends since high school, believed in me before I even believed in myself. And last, but certainly not least, my wife Kasey Cannon. Kasey has seen my best and my worst, and despite the latter has stuck by me for nearly eleven years at this point. She’s always been there to listen to my nervous breakdowns in the middle of the night, first when I couldn’t figure out homework problems that were due in the morning, and then later when I couldn’t get my code to work. My dissertation would have been unimaginably more difficult without her love and support. The material in this dissertation is based on work funded by the National Science Foundation under grants ACI-1642391, OAC-1841480, and PHY-1454389. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. The material in this dissertation is also based on work funded by the Pennsylvania State University’s academic computing fellowship.

xxvi Dedication

This dissertation is dedicated to my maternal grandmother, Patricia Messick, who has been my best friend for as long as I can remember.

xxvii Chapter 1 | Introduction to Gravitational- Wave Astronomy

While Newton was the first to successfully propose a mathematical framework to describe gravity, he was also the first to admit he did not understand why masses are gravitationally attracted to each other. This question remained unanswered until 1916, when Albert Einstein published his general theory of relativity, commonly referred to as general relativity. Einstein’s theory takes the concept of space-time from special relativity, that space and time are not separate entities but rather different components of one 4-dimensional object, and expands on it by allowing space-time to be curved by mass and energy. General relativity tells us that gravity is a consequence of the curvature of space-time and not actually a force. The curvature of space-time is described mathematically using an object called the metric tensor, gµν, or the metric for short; the metric in a given region of space-time is related to the distribution of mass and energy in that region via a set of differential equations known as Einstein’s field equations. The metric encodes how to compute distances in curved space-times, which means that both physical distances and the rate at which time passes change as gravity changes. For example, time runs marginally faster at high altitudes than at sea-level due to the decreased strength of the gravitational field at high altitudes. Furthermore, Einstein’s field equations can be shown to reduce to a wave equation, meaning that general relativity predicts the existence of gravitational waves, which cause physical distances perpendicular to their direction of travel to change. Although this will not be discussed in greater detail until Sec. 1.2, the most common sources

1 of gravitational-waves are compact binary mergers. Compact binaries are binary star systems comprised of compact objects, such as white dwarves, neutron stars, or black holes. This dissertation will focus primarily on neutron stars and black holes. Compact binaries slowly lose energy to gravitational waves, which causes their orbital separation to decrease and their angular frequency to increase, which in turn causes the rate at which they lose energy to increase and thus increases the rate at which their orbital separation and angular velocity change. If the orbital separation of a compact binary is small enough, it will radiate enough energy away via gravitational waves to merge within the lifetime of the universe. The first such system observed was the binary pulsar PSR B1913+16 [15], often referred to as the Hulse-Taylor binary after its discoverers, which will merge in 300 million years [16]. Pulsars are a type of neutron star, thus the discovery of the Hulse-Taylor binary proved that compact binaries that include at least one neutron star component do occur in small enough orbits to merge within the lifetime of the universe, which implies the existence of compact binaries with neutron star components that may be merging now or in the near future. This was especially exciting because the energetics believed to occur during binary neutron star (BNS) mergers and the rate at which they were estimated to occur made BNS mergers strong candidates for the origin of extremely energetic electromagnetic bursts known as short Gamma-Ray Bursts (GRBs), making BNS mergers a multi-messenger astronomical phenomenon. Multi-messenger astronomy is the study of astronomical phenomena that emit “messengers” across different channels, e.g. the sun emits photons and neutrinos. The first extra-galactic multi-messenger observation was SN1987A, a supernova observed via light and neutrinos in 1987 [17]. BNS mergers were a tantalizing multi-messenger candidate because it was believed that they emit gravitational waves up until merger, after which they emit an extremely collimated short GRB and an isotropic optical transient [18]. The short GRB only lasts ∼ 0.1 − 1 s, but the optical afterglow was believed to persist for hours to days, followed by an isotropic radio afterglow visible weeks to months after the merger [18]. The optical transient, commonly referred to as a kilonova, was expected to last about a day and to be followed by a radio afterglow years after the merger [18]. Until 2013, it was believed that observers could not see both the GRB and the kilonova, as the short

2 GRB was thought to only be visible if the observer was on-axis, meaning they are within the GRB’s small viewing angle. In 2013, a possible kilonova was observed alongside a short GRB, though the observation was not significant enough to claim a detection [19]. Detecting gravitational waves from a BNS merger in coincidence with a short GRB would confirm BNS mergers as short GRB progenitors, but rapidly detecting the gravitational waves would also give astronomers the ability to search for additional counterparts even in the absence of an observed short GRB. My dissertation research was focused on detecting gravitational waves from compact binary mergers in low latency with the intent of distributing alerts to other astronomers for multi-messenger follow-up. My research led to the first and, as of the writing of this dissertation, only multi-messenger BNS merger detection to date, a low-latency joint detection of gravitational waves and a short GRB that led to the detection of an accompanying kilonova and other counterparts. The first chapter of this dissertation will cover background material, including an introduction to general relativity, gravitational waves, ground-based interferometric gravitational-wave detectors, general compact binary merger gravitational wave data analysis methods, and the low-latency gravitational-wave analysis I co-developed, the GstLAL-based inspiral pipeline; chapters 2 through 4 will discuss results from the GstLAL-based inspiral pipeline for the first three confident gravitational-wave detections; chapter 5 will discuss the analysis of the first gravitational-waves observed by three ground-based detectors instead of two; chapter 6 will discuss the multi-messenger BNS detection; chapter 7 will discuss GstLAL results used in a catalog of gravitational wave detections made from 2015 to 2017; and chapter 8 will discuss detections made up until Spring 2019 and state of the field of gravitational-wave astronomy at the time of this dissertation’s writing.

1.1 Einstein’s General Theory of Relativity

1.1.1 Deﬁnitions

In general relativity, space-time is a manifold equipped with a metric, gµν, which describes how to perform inner-products on the manifold. The mathematical definition of a manifold is beyond the scope of this dissertation, but conceptually a

3 manifold can be thought of as a geometric object that has a one-to-one and onto mapping to Rn, though the same mapping isn’t necessarily the same all over the manifold, but rather in patches that cover the entire manifold. A common example is the 2-sphere, where every point can be mapped to Rn in at least one of two patches:

1. setting the origin at the north pole and covering every point on the sphere except the south pole,

2. setting the origin at the south pole and covering every point on the sphere except the north pole.

Vectors must be defined specifically for manifolds, as the common Euclidean definition of a vector won’t have the properties of a vector space on a curved surface [20]. On a manifold, vectors are defined as tangent to some smooth curve within the manifold at a point; the set of vectors tangent to the smooth curve form a vector-field. If the curve is parameterized by λ, the vector-field is defined as

d V = = V µeˆ , (1.1) dλ µ where we will generally use µ ∈ {0, 1, 2, 3},

∂xµ V µ = (1.2) ∂λ are the vector components, ∂ eˆ = (1.3) µ ∂xµ are the basis vectors, xµ = xµ(λ) is a four-vector that describes the curve in the chosen coordinate-space, and the Einstein summation convention is being used. The Einstein summation convention is that repeated indices are summed over, for example Eq. 1.1 could also be written as

3 X µ V = V eˆµ. (1.4) µ=0

In this dissertation, Greek letters will generally be used when the indices cover both space and time dimensions, with 0 as the time dimension, while Roman letters will generally be used for indices that only cover the spatial dimensions.

4 When performing a change of coordinates from xµ to x0µ, vectors transform as

∂x0µ V 0µ = V ν, xµ → x0µ. (1.5) ∂xν

Objects that obey the transformation law Eq. 1.5 are said to be contravariant, and are written with a superscript index. In addition to defining vectors, we need to define dual vectors, sometimes referred to as one-forms or covectors. A dual vector is an object that acts on a vector and returns a real number or returns a real number when acted on by a vector. An example for readers familiar with quantum mechanics is the “bra” in a bra-ket multiplication; in other words, if you have a state α represented by the vector |αi, the state’s dual vector is hα|. It is important to note that dual vector spaces are defined with respect to specific vector spaces, just as an arbitrary bra hβ| won’t produce a real number when acted on |αi. Dual vectors transform as

∂xν V 0 = V , xµ → x0µ. (1.6) µ ∂x0µ ν

Objects that obey the transformation law Eq. 1.6 are said to be covariant, and are written with a subscript index. For more details on dual vectors, see Refs. [20,21]. The concept of tensors can be extended from the conversation of vectors and dual vectors. Once again, the mathematical details can be found in Refs. [20, 21], but broadly speaking a tensor of type (k, `) is a mapping from k vectors and ` dual vectors to Rn. In other words, vectors are tensors of type (1, 0), while dual vectors are tensors of (0, 1). A tensor of type (k, `) will have k superscript indices and ` subscript indices. Tensors of type (k, `) transform as

∂x0µ1 ∂x0µk ∂xβ1 ∂xβ` 0µ1···µk α1···αk µ 0µ T ν ···ν = ··· ··· T β ···β , x → x (1.7) 1 ` ∂xα1 ∂xαk ∂x0ν1 ∂x0ν` 1 `

Notice that this is a generalized version of Eqs. 1.5 and 1.6. In fact, the (k, 0) case is the generalized transformation law for contravariant objects with k superscripts, and the (0, `) case is the generalized transformation law for covariant objects with ` subscripts. The careful reader may notice that while vectors were deﬁned mathematically as tangent to a curve, the dual vector was not deﬁned. That is because it is derived from a vector; that is, we need a tensor that will take a vector and return

5 a dual vector to deﬁne the dual vector. The tensor that is used to accomplish this is known as the metric, gµν. It is symmetric, gµν = gνµ, and non-degenerate, µν det |gµν|= 6 0. The inverse metric, g , is deﬁned such that

µα α gµνg = δν , (1.8)

α where δν = 1 if α = ν and 0 otherwise. The metric is used to raise and lower indices of tensors, which allows us to ﬁnally deﬁne the dual vector,

ν Vµ = gµνV . (1.9)

The metric is also used to deﬁne the line element, ds2, on a manifold,

2 µ ν ds = gµνdx dx . (1.10)

Eq. 1.10 allows us to deﬁne a physically meaningful quantity on a manifold, while the coordinates themselves often have no physical interpretation. For example, if we treat the Earth as a perfect sphere, we can compute the distance travelled, s, by somebody who travelled along a curve parameterized by λ from λ1 = (RE, φ1, θ1) 1 to λ2 = (Re, φ2, θ2) , where RE is the radius of the earth,

s Z q Z λ2 i j ν µ 0 ∂x ∂x s = gijdx dx = dλ gij 0 0 . (1.11) λ1 ∂λ ∂λ

For simplicity, we will assume the path being walked is at the equator from φ1 = 0 to φ2 = π. We can describe this curve as

1 x = RE, (1.12) x2 = λ, (1.13) π x3 = . (1.14) 2 1Note that Roman letters are used in Eq. 1.11 to denote that we are only considering spatial dimensions.

6 Now using the metric for spherical coordinates (r, φ, θ),

  1 0 0    2 2  gij = 0 r sin θ 0  , (1.15)   0 0 r2

Eq. 1.11 becomes

v Z π u 2 !2 u 2 2 π ∂x s = dλ tRE sin = Reπ (1.16) 0 2 ∂λ as expected.

1.1.2 Curvature

Unlike Euclidean systems, a manifold can have intrinsic curvature. The eﬀect of this curvature is measured by parallel transporting a vector along a closed path on the manifold, where parallel transport is the act of displacing a vector while ensuring it is parallel to itself before and after each inﬁnitesimal step. In a Euclidean coordinate system, if you parallel transport a vector along a closed path then it will retain its original orientation when it returns to the starting point, but this is not in general true on a manifold. Consider a curve on our manifold parameterized by λ, we already know from Eq. 1.2 that in some coordi- µ ∂xµ nate system the vector components tangent to that curve are V = ∂λ . Following Ref. [22], we seek the collection of all vectors, U µ, that V µ can be parallel trans- ported to. In other words, we seek the collection of vectors that do not change along the curve parameter, λ,

dU µ ∂xν 0 = = ∇ U µ = V ν∇ U µ, (1.17) dλ ∂λ ν ν where ∇ν is the covariant derivative operator. In the case of no curvature, the ∂ derivative operator is what we expect in Euclidean geometry, ∇ν = ∂ν ≡ ∂xν . Notice that the derivative with respect to a contravariant component, xν, is itself ∂ covariant (meaning ∂ν ≡ ∂xν transforms according to Eq. 1.6). Eq. 1.17 describes the parallel transport of the vector U µ along the curve parameterized by λ (re- µ ∂xµ member: V = ∂λ ).

7 Applying the change of variables transformations laws, Eq. 1.7 [22], to Eq. 1.17,

µ µ µ β ∇νU = ∂νU + ΓαβU , (1.18) ρ ∇νUµ = ∂νUµ − ΓνµUρ, (1.19) ∂xρ ∂2x0δ 1 Γρ ≡ = gρσ(∂ g + ∂ g − ∂ g ) (1.20) µν ∂x0δ ∂xµ∂xν 2 µ νσ ν σµ σ µν where Eq.1.20 are the Christoffel symbols [20,22]. We now have all of the tools necessary to describe intrinsic curvature. The Riemann curvature tensor is defined as the difference between parallel transporting a vector first one direction then another direction and parallel transporting it the latter direction first and then the former direction [22],

ρ σ ρ Rµνσ V = − (∇µ∇ν − ∇ν∇µ) V . (1.21)

Conceptually, the Riemann curvature tensor encodes how much the space-time we’re considering differs from a flat space-time (a space-time with no curvature). The Riemann curvature tensor is often written in terms of the Christoffel symbols,

ρ ρ ρ ρ δ ρ δ Rµνσ = −∂µΓνσ + ∂νΓµσ − ΓµδΓνσ + ΓνδΓµσ. (1.22)

1.1.3 Einstein’s Field Equations

We can now write down Einstein’s ﬁeld equations, which describe the interaction between space-time, represented mathematically by the metric tensor gµν, and matter/energy, represented mathematically by the stress-energy tensor T µν. The derivation of Einstein’s equations is beyond the scope of this dissertation, but we will show that they reduce to the Newtonian physics that we expect in the appropriate limit. Einstein’s ﬁeld equations are

1 8πG R − g R = T , (1.23) µν 2 µν c4 µν where

δ Rµν = Rµδν , (1.24) µ R = R µ, (1.25)

8 are referred to as the Ricci curvature tensor and the Ricci curvature scalar respectively.

1.1.3.1 The Newtonian Limit

In this section, we will show that Einstein’s ﬁeld equations (Eq. 1.23) reduce to the Newtonian description of gravity in the limit that gravity is weak, matter is slow moving, and internal stresses in matter are small. Here we will quote results from, and at times loosely follow, the derivation in Ref. [22]. In the Newtonian limit, we assume space-time is barely curved; physically, we’re assuming that gravity is weak. In this limit, the metric can be described as the

ﬂat-space-time metric, ηµν, with a small perturbation, hµν,

gµν = ηµν + hµν. (1.26)

The ﬂat-space-time metric, ηµν, is known as the Minkowski metric. This dissertation uses the Minkowski coordinates xµ = (t, x, y, z) and the (−, +, +, +) convention, thus the Minkowski metric is

  −c2 0 0 0      0 1 0 0 η =   . (1.27) µν    0 0 1 0   0 0 0 1

µν The inverse metric, g , is obtained by Taylor expanding the inverse of gµν around hµν, gµν ≈ ηµν − hµν. (1.28)

In this approximation, we can use the Minkowski metric as the raising and lowering operator to first order instead of the complete metric. The result of plugging Eq. 1.26 into the Einstein field equations (Eq. 1.23) can be simplified by defining the trace-reversed metric perturbation,

1 h¯ = h − η h. (1.29) µν µν 2 µν

9 ¯ ¯µ 2 Notice h = h µ = −h, hence the name trace-reversed metric perturbation . The result can be further simpliﬁed by using the Lorenz gauge, deﬁned as a coordinate system where ∂h¯µν = 0. (1.30) ∂xµ

Dropping all higher order terms of hµν, the ﬁeld equations yield the wave-equation,

16πG h¯ ≈ − T , (1.31) µν c4 µν where is the ﬂat space-time d’Alembertian operator. We must now write down the stress-energy tensor, T µν, in the Newtonian limit. In general, the time-time component of the stress-energy tensor, T 00, is the relativistic-mass density; the space-time components, T 0i = T i0, represent the relativistic mass ﬂux through a surface xi; and the space-space components, T ij, represent the internal stresses. The Newtonian limit assumes small velocities and internal stresses, which means that the only non-zero component of the stress-energy tensor is the time-time component, T 00 = ρ, where ρ is the mass density3. We can also assume that the time derivatives of the metric perturbation in Eq. 1.31 are small compared to the spatial derivatives, because slow motion implies the metric is slowly varying4. Using these assumptions, Eq. 1.31 becomes

2¯ ∇ h00 = −16πGρ, (1.32) 2¯ ∇ h0i = 0, (1.33) 2¯ ∇ hij = 0. (1.34)

Comparing Eq. 1.32 to the Poisson equation for Newtonian gravity, we see that if

¯ h00 = −4Φ, (1.35) where Φ is the Newtonian gravitational potential, then Eq. 1.32 becomes the New-

2 µν µ µν The trace of a tensor T is deﬁned as T ≡ T µ = gµν T . 3The relativistic energy is the rest energy to leading order at small velocities. 4If the sources of small changes in the metric are slow, then it is reasonable to assume the changes to the metric are also slow.

10 tonian gravity Poisson equation,

∇2Φ = 4πGρ, (1.36) and the Newtonian gravitational ﬁeld, ~g, is given by

1 ~g = −∇Φ = ∇h¯ . (1.37) 4 00 thus general relativity reproduces Newtonian gravity in the Newtonian regime.

Using the trivial solutions for Eqs. 1.33 and 1.34, the Newtonian metric gµν is

  −c2 − 2Φ 0 0 0    2Φ   0 1 − 2 0 0  g =  c  , (1.38) µν  2Φ   0 0 1 − c2 0   2Φ  0 0 0 1 − c2 to leading order.

1.2 Gravitational Waves

In 1916, Einstein realized that the wave equation could be derived from his field equations in the limit of nearly flat space-time. Consider the vacuum case, where T µν = 0. In this case, the solution to Eq. 1.31 is a plane wave. Einstein was the first to derive these plane wave solutions to his linearized equations [23]. In this section, I will derive the leading order approximation of the gravitational-wave amplitude far away from the source, following the derivation in Ref. [22]. The leading order approximation is known as the “quadrupole formula”, though that phrase can also refer to a first order approximation of the gravitational wave luminosity; both will be discussed here.

1.2.1 Quadrupole Formula

Mathematically describing gravitational waves is easiest in the weak-field regime. This section will loosely follow the derivation in [22]. We will start with Einstein’s field equations in the weak-field, or “linear gravity”, regime, derived in Sec. 1.1.3.1

11 and shown in Eq. 1.31. As before, the Minkowski metric, Eq. 1.27, is used for raising and lowering indices instead of the full metric. In order to solve Eq. 1.31, we need the retarded Green’s function for the d’Alembertian operator, which we can treat as the ﬂat-space-time d’Alembertian operator to ﬁrst order in h. The retarded Green’s function is then

1 x G(t, ~x) = δ t − Θ(t), (1.39) 4πx c where Θ(t) is the Heaviside-step function. Thus the solution to Eq. 1.31 is5

16πG Z h¯µν(t, ~x) ≈ − d4x0 G (t − t0, ~x − ~x0) T µν(t0, ~x0) c4 |~x−~x0| 4G Z T µν t − , ~x0 ≈ − d3x0 c , (1.40) c4 |~x − ~x0|

In the far-zone regime, the distance to the source is much larger than the wavelength of the gravitational wave, which is itself much larger than the characteristic size of the source,

r λ Rsource, (1.41) thus |~x − ~x0| ≈ r in Eq. 1.40 and

4G Z r h¯µν(t, ~x) ≈ − d3x0 T µν t − , ~x0 . (1.42) rc4 c

The Lorenz gauge allows us to solve for the spatial components of the perturbation in isolation, as the gauge conditions (Eq. 1.30) will then give us the time µν components. In the Lorenz gauge, ∂µT = 0, which provides us the following identity [22],

1 ∂2 ∂ 1 ∂2 T ij = xixjT 00 + xiT jk + xjT ki − xixjT k` . (1.43) 2 ∂x02 ∂xk 2 ∂xk∂x` Plugging this into the spatial components of Eq. 1.42 and dropping boundary terms, 2G r hij(t, ~x) ≈ I¨ij(t − , ~x), (1.44) rc4 c 5The indices at this point are sometimes given as superscripts and sometimes as subscripts. As long as one is consistent throughout their derivation, it is ﬁne to use either and to switch back by using the metric to raise or lower the indices.

12 where Z Iij(t, ~x) = d3x0 x0ix0jT 00 (t, ~x0) (1.45) is the quadrupole moment tensor. It is common in the literature to perform another gauge transformation at this ¯µν ¯µν point. This is only possible outside of the source, where h = 0. If h 6= 0, then T µν takes up the degrees of freedom that allow us to perform this transformation [24]. These degrees of freedom allow us to choose a Lorenz gauge where the metric perturbation is spatial and traceless [20,22,24,25], leaving us with only two degrees of freedom: the two gravitational-wave polarizations, h+ and h×. This sub-gauge is called the transverse-traceless gauge, and is denoted by a subscript or superscript TT. We can now project this solution into the transverse-traceless gauge with the projection operator,

Pij = δij − nˆinˆj, (1.46)

~x where nˆ = r is the unit vector in the direction of propagation.

2G r hij (t, ~x) ≈ I¨ij (t − , ~x), (1.47) TT rc4 TT c where

1 Iij = P ikI P `j − P ijP k`I . (1.48) TT k` 2 k`

Eq. 1.47 is known as the quadrupole formula. Although Eq. 1.47 is often referred to as the quadrupole formula in the literature, the equation describing gravitational-wave luminosity is also often referred to as the quadrupole formula. Deriving the luminosity formula requires extending the linear approximation to quadratic order in hµν. Refs [20,22,24] are good references for readers looking for careful derivations of the following statements. We will now consider the second order metric

(2) gµν = ηµν + hµν + hµν . (1.49)

This will generate two new types of terms in the Einstein ﬁeld equations: prod- (2) ucts of hµν that were discarded in the linear approximation and derivatives of hµν .

13 (2) These can be rearranged to form a second order Einstein field equation for hµν , (2) where the stress-energy tensor is given by the products of hµν; in other words, hµν carries information about the curvature induced by the first order metric perturbation, c4 T GW = h∂ hij ∂ hTTi, (1.50) µν 32πG µ TT ν ij where h· · · i denotes a spatial average over many oscillations. To compute the luminosity, we want to integrate the gravitational-wave flux,

d2E GW = T 03 , (1.51) dAdt GW over a sphere around the source in the near-zone regime. The near-zone regime is the regime where the wavelength of the gravitational wave is much larger than the distance to the source which is itself much larger than the characteristic size of the source,

λ r Rsource. (1.52)

The near-zone solution to Eq. 1.31 is almost identical to Eq. 1.44, except it has the reduced quadrupole moment tensor,

Z 1 I-ij(t, ~x) = d3x0 (x0ix0j − |~x0|2δij)T 00(t, ~x), (1.53) 3 in place of Iij(t, ~x). Notice that I-ij is identical to Iij other than its vanishing trace, -ij ij thus ITT = ITT and Eq. 1.47 is valid in both the near-zone and far-zone. The result of integrating Eq. 1.51 over a sphere in the near-ﬁeld regime will not be derived here, merely quoted from Ref. [22],

dE G ...... L = − GW = h I- I- iji. (1.54) GW dt 5c5 ij

Eq. 1.54 can be used to estimate the change in angular momentum of the source over time by deﬁning a radiation reaction force which does work on the source to remove energy at a rate of LGW [22],

dJ i 2G ... = − ijkhI¨- I- ` i, (1.55) dt 5c5 j` k

14 where ijk is the Levi-Civita symbol.

1.2.1.1 Quadrupole Formula of Binary

For a binary comprised of point masses with masses m1 and m2 in a circular orbit in the xy plane with the spatial origin set at the binary’s center of mass,

00 T = m1δ(~x − ~x1) + m2δ(~x − ~x2), (1.56) where

~x1 = r1(cos(ωt)ˆx + sin(ωt)ˆy), (1.57)

~x2 = r2(cos(ωt + π)ˆx + sin(ωt + π)ˆy), (1.58)

are the positions of the binary components, mi is the mass of the ith component, ω is the orbital frequency, and ri is the distance from the center to the ith component. Choosing the z direction as the direction of propagation and writing the equations down in terms of the binary diameter, R = r1 + r2, and the reduced mass, µ = m1m2 , m1+m2

  cos(2ω(t − r )) sin(2ω(t − r ) 0 4GµR2ω2  c c  ij  r r  hTT(t, ~x) ≈ − 4 sin(2ω(t − c )) − cos(2ω(t − c )) 0 . (1.59) rc   0 0 0

There are two polarizations, named “h-plus”, h+, and “h-cross”, h×, are discernible from Eq. 1.59,

h(t, ~x) = h+(tr, ~x)ˆe+ + h×(tr, ~x)ˆe× (1.60) 4GµR2ω2 h (t, ~x) ≡ − cos(2ωt), (1.61) + rc4 4GµR2ω2 h (t, ~x) ≡ − sin(2ωt),. (1.62) × rc4

15 r where tr = t − c is the retarded time and

  1 0 0     eˆ+ = 0 −1 0 , (1.63)   0 0 0   0 1 0     eˆ× = 1 0 0 . (1.64)   0 0 0

To understand the names of the polarizations, consider a set of test particles in the form of a unit circle. We want to compute the distance between any two points on opposite sides of the circle with our metric. This is easiest to do in spherical coordinates; using Eq. 1.7, the rr term of the spatial metric in spherical coordinates (r, φ, θ) (notice the r here is the spherical coordinate r, not the distance to the source r) is found to be

2 grr = 1 + sin θ(h+(tr, ~x) cos(2φ) + h×(tr, ~x) sin(2φ)). (1.65)

Taking our circle of test particles to be in the z = 0 plane, we have θ = π/2,

Z Z 1 √ ds = 2 dr g across rr circle 0 q = 2 1 + (h+(tr, ~x) cos(2φ) + h×(tr, ~x) sin(2φ)). (1.66)

From Eq. 1.66, we see that distance from (x, y, z) = (−1, 0, 0) to (1, 0, 0) and the distance from (0, −1, 0) to (0, 1, 0) are

Z (1,0,0) q ds = 2 1 + h+(tr, ~x), (1.67) (−1,0,0) Z (0,1,0) q ds = 2 1 − h+(tr, ~x). (1.68) (0,−1,0)

In other words, the plus-polarization stretches one axis of the “+” while compressing the orthogonal axis. Similarly, the cross polarization stretches one axis of the

16 “×” while compressing the orthogonal axis,

√ √ 2 2 Z ( 2 , 2 ,0) q √ √ ds = 2 1 + h×(tr, ~x), (1.69) 2 2 (− 2 ,− 2 ,0) √ √ 2 2 Z ( 2 ,− 2 ,0) q √ √ ds = 2 1 − h×(tr, ~x). (1.70) 2 2 (− 2 , 2 ,0)

The luminosity of a gravitational-wave emitted from a compact binary is

32µ2R4ω6G L = , (1.71) GW 5c5 and the rate the binary loses angular momentum is

d|J~| dJ 3 32µ2R4ω5G = = − . (1.72) dt dt 5c5

1.3 Compact Binaries

The binary described above assumed the two orbiting bodies were point masses, a simpliﬁcation that does not approximate reality for most binaries. Binaries where both components are compact objects such as neutron stars or black holes are well described by this approximation however [26]. The ﬁrst such binary observed was PSR B1913+16 [15, 16], discovered in 1974 and now known as the Hulse- Taylor binary after its discoverers. After six years of observations, the Hulse- Taylor binary’s orbital period was found to be decreasing at a rate that matched the prediction by general relativity [16]. Meanwhile, astronomers were beginning to think about merging neutron star binaries as potential gravitational wave sources, though the estimated rates had several orders of magnitude uncertainty [27].

1.3.1 Merger Mechanism

To qualitatively understand how compact binaries react to the radiation of energy via gravitational waves, we can model the mechanism classically. For the binary described above, the energy at any given point in time classically is

1 Gm m E = µR2ω2 − 1 2 . (1.73) 2 R

17 In the Newtonian limit, we have shown that compact binaries lose energy at a rate given by Eq. 1.71. In Newtonian mechanics, we model all of the eﬀects of space- time (gravity) as a potential energy, so the energy lost via gravitational waves will come from the gravitational potential energy; in other words, if a binary has gravitational potential energy Ug at time t, then it will have gravitational potential energy Ug − LGWdt at time t + dt, where dt is an inﬁnitesimally small interval of time,

Gm m Gm m − 1 2 = − 1 2 − L dt, R(t + dt) R(t) GW L dt !−1 → R(t + dt) = R(t) 1 + GW . (1.74) Gm1m2/R

Since the separation of the binary decreases, the angular frequency increases,

1 Gm m 1 Gm m µR(t + dt)2ω(t + dt)2 − 1 2 = µR(t)2ω(t)2 − 1 2 − L dt, 2 R(t + dt) 2 R(t) GW L dt ! → ω(t + dt) = ω(t) 1 + GW . (1.75) Gm1m2/R(t)

Following Ref. [22], it is convenient to rewrite the gravitational-wave luminosity and amplitude in terms of the orbital velocity, v = Rω. We can use Kepler’s third law to show that ν is inversely proportional to R and directly proportional to ω,

s GM v = Rω = (GMω)1/3 = , (1.76) R where M = m1 + m2 is the total mass, thus writing everything in terms of the orbital velocity allows us to only deal with one variable that changes in time instead of multiple. In terms of v, the energy and luminosity are given by

1 E = − µv2, (1.77) 2 32η2c5 v 10 L = , (1.78) GW 5G c

µ m1m2 dE where η = M = M 2 is the symmetric mass ratio. From LGW = − dt , we can show dv 32ηc4 v 9 = . (1.79) dt 5GM c

18 Eq. 1.79 can be integrated and solved for t to ﬁnd the time until coalescence, or merger, tc 5GM v −8 t = 0 , (1.80) c 256ηc3 c where v0 is the original orbital velocity. Eq. 1.79 can also be used to solve for the gravitational-wave frequency evolution, where the gravitational-wave frequency f = 2forbital = ω/π,

df df dv 96π8/3 (GM)5/3 f 11/3 = = , (1.81) dt dv dt 5c5

3/5 3/5 (m1m2) where M = η M = 1/5 is called the chirp mass. It is called the chirp mass (m1+m2) because it is the only combination of masses that aﬀect the frequency evolution to leading order, which “chirps” in the last moments before merger. We will see that it is also the only combination of masses that aﬀect the gravitational-wave amplitude to leading order. A natural form to write the gravitational-wave amplitudes, Eqs. 1.61 and 1.62, in is one where time is relative to the coalescence time. In order to do this, we must rewrite the amplitudes in term of the orbital velocity, v, solve for time and the orbital phase relative to their coalescence values as functions of orbital velocity, and dE then replace the orbital velocity using these functions. Starting with LGW = − dt , we can show dt 5GM v −9 = . (1.82) dv 32c4η c

We can use this to express time relative to tc,

Z vc 10 0 dt 5GM v t(v) = tc − dv = tc − , (1.83) v dv0 256ηc3 c where vc is the orbital velocity at the moment of coalescence and the integral is done by taking vc → ∞. We can use a similar trick to solve for the phase of the binary in terms of the coalescence phase φc,

Z vc −5 0 dφ 1 v φ(v) = φc − dv = φc − , (1.84) v dv0 32η c

dφ dφ dt dt where dv = dt dv = ω dv , and again the integral is done by taking vc → ∞. We can now solve for v as a function of t and tc using Eq. 1.83, and plug this and Eq. 1.84

19 into Eqs. 1.61 and 1.62 to get

3 !−1/4  3 !5/8 GM c (tc − tr) c (tc − tr) h+(t, ~x) = − cos 2φc − 2  , (1.85) c2r 5GM 5GM

3 !−1/4  3 !5/8 GM c (tc − tr) c (tc − tr) h×(t, ~x) = − sin 2φc − 2  . (1.86) c2r 5GM 5GM

Now we can see that the gravitational-wave amplitude increases as the binary approaches the merger time. We can also see that compact binaries with larger chirp masses emit gravitational-waves with larger amplitudes. So far we have assumed that the orbit is in the xy-plane and the gravitational wave is traveling in the zˆ direction, if it is instead traveling at an angle ι to zˆ, then instead [22]

2 3 !−1/4  3 !5/8 GM 1 + cos ι c (tc − tr) c (tc − tr) h+(t, ~x) = − cos 2φc − 2  , c2r 2 5GM 5GM (1.87)

3 !−1/4  3 !5/8 GM c (tc − tr) c (tc − tr) h×(t, ~x) = − cos ι sin 2φc − 2  . (1.88) c2r 5GM 5GM

The life of a compact binary can be thought of as having three stages. The stage described so far is referred to as the inspiral phase [22]. Towards the end of the inspiral stage, the linear gravity approximation becomes invalid. This late stage of the inspiral and the merger itself are referred to as the merger stage, and during this stage Einstein’s equations can only be solved numerically. Finally, after the merger the resulting body must radiate away excess energy, which it does via quasi-normal modes during the ringdown stage [22]. These will be discussed more in Sec. 1.3.3.

1.3.2 Merger Rates

The early attempts to estimate binary neutron star (BNS) rates after the discovery of the Hulse-Taylor binary concluded that binary neutron stars that are close enough to coalesce would not form at a high enough rate to be a strong gravitational wave progenitor candidate [28]. In 1979, Ref. [29] estimated a much higher

20 rate using the Hulse-Taylor binary by assuming that the number of pulsars in binary systems is Poisson distributed and that binary neutron stars are the end result of stellar evolution for massive X-ray binaries. This calculation estimated a merger rate of 2.9 ± 1.6 × 10−4 yr−1 in the Milky Way galaxy. Ref. [30], a summary of the state of compact binary merger rate estimates in 2010, refers to this type of estimate as having units of MWEG−1 yr−1, where MWEG stands for Milky way equivalent galaxy. The unit typically used when discussing compact binary merger rate estimates in modern literature is Gpc−3 yr−1. Ref. [30] provides the 7 MWEG 1.98×10 L10 conversion factor 3 = 1, though notes that there is a large amount 1.7L10 Gpc of uncertainty in the conversion factor and that blindly applying the conversion ignores non-spiral galaxies and thus underestimates the rate [30]. Using this conversion factor, the rate estimate in Ref. [29] is 3.4 ± 1.9 × 103 Gpc−3 yr−1. 10 years later, Ref. [27] updated this estimate to include several uncertainties, e.g. the low number of observed binary neutron stars (which was still only the Hulse-Taylor binary at this time). With uncertainties, the new rate estimate was 10−2 −102 yr−1 out to 100 Mpc, which corresponds to a rate of 2 − 2 × 104 Gpc−3 yr−1. By 1991, more pulsar binaries6 had been discovered. Two seminal papers used a subset of these to estimate the binary neutron star merger rate; both papers improved upon previous estimates by estimating and incorporating the fraction of the volume of space that had been probed with pulsar surveys. Ref. [31] used pulsar binaries whose orbits were less than half a day, two of which were in our galaxy and the other in a nearby globular cluster, to compute an ‘ultraconservative’ rate estimate and a best guess rate estimate. The ultraconservative rate estimate was computed to be 3 yr−1 within 1 Gpc, or 0.7 Gpc−3 yr−1, while their best guess was 3 yr−1 within 200 Mpc, or 90 Gpc−3yr−1. Ref. [32] used the same three pulsar binaries, along with another that had an orbital period greater than Ref. [31]’s cutoﬀ of half a day, to estimate a rate of 1 yr−3 within 200/h Mpc, where h is the dimensionless Hubble parameter. Using the 2018 Planck result −1 −1 −3 −1 of H0 = 67.7 km s Mpc [33], this is a merger rate density of 9 Gpc yr . Both of these papers use diﬀerent methods to estimate the rate, both of which involve many uncertainties that they discuss, so determining rates within an order of magnitude of each other is impressive.

6Pulsar binaries refers to a binary system that contains at least one pulsar, not necessarily a binary where both objects are pulsars.

21 In 2003, a third galactic binary pulsar was discovered that had the smallest orbital period yet at 2.4 hours [34,35]. Ref. [35] used a new method that simulated selection effects and performed a Bayesian analysis7 to estimate a probability distribution for the merger rate, allowing them to quote a confidence interval (e.g. it is within this range with 90% confidence) as opposed to quoting a point estimate with an uncertainty. This new method also allowed them to estimate the merger rate with different population models. Their preferred pulsar model, based on observations at the time, predicted the most likely merger rate and its 95% confi- +209.1 −6 −1 −1 +2434 −3 −1 dence interval were 83.0−66.1 × 10 MWEG yr , or 966−769 Gpc yr . The 2010 summary of estimated compact binary rates uses the most likely value from this model as its “realistic” estimate of merger rates, though it rounds to the value to 1 significant digit [30]. The summary then quotes the lowest value across all of the published models in Ref. [35] as the most pessimistic estimate and the highest value across all of the models as the most optimistic prediction. In modern units and to 1 significant figure, these are 10 Gpc−3 yr−1 and 104 Gpc−3 yr−1. At sensitivities expected for advanced ground-based detectors, these estimated become 0.4, 40, and 400 detections per year. This was the first merger rate estimate that predicted the rate to likely be high enough to offer prime candidates for gravitational wave detections. Many papers attempted to estimate rates for neutron star black hole binary (NSBH) mergers and binary black hole mergers, however these estimates were nearly free of observational constrains and thus had much larger uncertainties because these systems had never been observed. The 2010 summary of merger rates [30] quotes NSBH merger rate estimates derived from a population synthesis model that was tuned to produce the observed rate of compact binaries that include neutron stars and the observed core-collapse supernova rate [36]. The pessimistic, realistic, and optimistic estimates were 0.6 Gpc−3 yr−1, 30 Gpc−3 yr−1, and 103 Gpc−3 yr−1. The synthesis models used to predict the NSBH merger rates were not used to make predictions for binary black hole merger rates because it did not account for galaxies that could possibly contribute significantly to black hole black hole merger rates but not the merger rate of binaries that include a neutron star. Instead, the 2010 summary [30] used an older publication [37] from the same group that did estimate binary black hole merger rates even though it didn’t

7The term “Bayesian analysis” will be deﬁned in a later section.

22 include the galaxies in question either. The pessimistic, realistic, and optimistic estimates were 10−1 Gpc−3 yr−1, 5 Gpc−3 yr−1, and 300 Gpc−3 yr−1.

1.3.3 Waveform Approximations

The waveform derived in Sec. 1.3.1 for gravitational waves from merging compact binaries, Eqs. 1.85 and 1.86, are only the leading order approximations for waveforms during a merging compact binary’s inspiral stage. The approximation can be expanded in powers of v/c in an approximation called the Post-Newtonian approximation. The post-Newtonian approximation assumes the binary components are compact enough to treat as point particles, an assumption that holds for binaries containing neutron stars and/or black holes and with a total mass less than

20M . The assumption can hold for compact binaries with larger total masses as well, but if the symmetric mass ratio is too small then the point mass assumption begins to fail. Though it will not be derived here, phase in the post-Newtonian expansion is affected by the ratio of masses to first order. At higher orders, the phase is also affected by spin-spin interactions between the two components and spin-orbit interactions between the spin of the components and the orbital angular momentum. The component spins interact with the orbital angular momentum such that the orbital plane will precess if the component spins are not parallel to the orbital angular momentum, adding an additional modulation to the waveforms [22]. The derivation of the first order post-Newtonian expansion is similar to the derivation of Eqs. 1.85 and 1.86, but it requires the definition of a new surrogate variable in place of the orbital velocity. Due to the increased complexity however, there are many different ways of solving the differential equations that model the binary’s evolution as a function of time [22]. There are thus different algorithms to compute the post-Newtonian approximations for different compact binary systems. These algorithms are often referred to as waveform approximates. For example, the TaylorT1 and TaylorT2 approximates are two different implementation of the post-Newtonian expansion, which compute waveforms in the time domain as was done here [22,38]. Other approximates compute the waveform in the frequency domain, such as the TaylorF2 approximate [38]. Higher orders of the post-Newtonian approximation can also be solved in different ways, thus there are in general many

23 different waveform approximates that can be used to model the inspiral stage of compact binaries. Many of these approximates assume that the component spins are aligned (or anti-aligned) with the orbital angular momentum, as the precession induced by non-aligned spins is an additional complexity. Compact binaries with small symmetric mass ratios require other methods to model the inspiral phase, for example effective-one-body formalisms that map the two-body problem to a test particle orbiting a black hole larger than either of the original components [39–41], or phenomenological models that fit many tunable parameters to numerical solutions [42]. As stated above, the merger stage requires numerical solutions, however the end of the inspiral stage can combined with numerical solutions for the merger stage via interpolation [22]. The ringdown stage can be described analytically. Waveform approximations which include all three stages are often referred to as IMR waveforms.

1.4 The LIGO Detectors

Early gravitational-wave detection efforts focused on resonant bar detectors that are sensitive to gravitational waves in a narrow band near their natural frequency [22]. Despite an early detection claim in the late 1960s [43], these efforts did not pan out. In this section I will focus on ground-based interferometric detectors, which at the time of writing are the only type to have made confident detections. Interferometric detectors, such as LIGO and Virgo, use a beam splitter to split a beam of light into two, which are then sent down the perpendicular arms of the detectors (Fig. 1.1). At the end of the arms are mirrors that reflect the beams back to the splitter. When the two beams recombine, if they have traveled the same length (or have differed by an integer number of wavelengths) then the light going back towards where it came from (the symmetric output) will constructively interfere while the light going the other direction (towards the photodiode in Fig. 1.1, known as the antisymmetric output) will completely destructively interfere, but if not then there will be light at the antisymmetric output [22]. The antisymmetric output can thus be used to infer if the lengths of the detector arms have changed. Nothing in the interferometer detector description so far differentiates it from a Michelson interferometer, but naive Michelson interferometers are not sensitive

24 Figure 1.1. A schematic of the basic interferometer design of ground-based interferometric detectors. Naively, the detector can be thought of as a Michelson interferometer, though a naive Michelson interferometer would not be sensitive enough to detect gravitational waves. This figure originally appeared in Ref. [1]. enough to detect gravitational waves even with arm lengths of O(1) km [22]. Mod- ern ground-based interferometric detectors are extremely sophisticated instruments and a thorough overview is outside of the scope of this dissertation. However, it is worth mentioning that both the LIGO and Virgo detectors were both designed with an initial stage and an advanced stage in mind. Initial LIGO took data from 2002 to 2010, and then shut down for upgrades. The upgraded version, advanced LIGO, began taking data in 2015. Initial Virgo took data from 2007 to 2011 before shutting down for upgrades. The upgraded version, advanced Virgo, began taking data in 2017. Now I’ll discuss how the passing of a gravitational wave affects an interferometer. Gravitational waves are tensor fields, so in order to understand how a detector will respond to a gravitational wave, the tensor representing the metric perturba- ij tion h must be projected onto the detector tensor, Dij, to get the induced strain, δL ij L = h Dij. To understand the detector’s response to any possible signal, we describe the natural coordinate system of the gravitational wave in terms of the natural coordinate system of the detector. We will use the arms of the detector as the x and y axes in the detector frame, such that the coordinate system is right-handed. The detector tensor is then

1 D = (ˆx ⊗ xˆ − yˆ ⊗ yˆ) , 2

25   1 0 0 1     = 0 −1 0 , (1.89) 2   0 0 0 and the gravitational wave signal we want to project onto the detector is

h(t, ~x) = h+(tr, ~x)ˆe+ + h×(tr, ~x)ˆe×, (1.90) where

GW GW GW GW eˆ+ =e ˆx ⊗ eˆx − eˆy ⊗ eˆy (1.91) GW GW GW GW eˆ× =e ˆx ⊗ eˆy +e ˆy ⊗ eˆx (1.92) are the polarization tensors in the signals natural coordinate system (which has unit GW vectors such as eˆx as opposed to xˆ). In the detector frame, if the gravitational wave is coming from an azimuthal angle φ and polar angle θ but the wave’s natural coordinate system is otherwise aligned with the detector’s, then

  cos θ cos φ   GW −1 −1   eˆx = Ry (θ)Rz (φ)ˆx =  − sin φ  , (1.93)   sin θ cos φ   cos θ sin φ   GW −1 −1   eˆy = Ry (θ)Rz (φ)ˆy =  cos φ  , (1.94)   sin θ sin φ

where Ri(α) is a rotation matrix for a rotation of α about axis ˆi. Performing this rotation and then plugging the result into the equations above, we ﬁnd

δL δL − δL = x y = F (θ, φ)h + F (θ, φ)h (1.95) L L + + × × where

1 F (θ, φ) = (1 + cos2 θ) cos(2φ), (1.96) + 2

F×(θ, φ) = cos θ sin(2φ) (1.97)

26 are the polarization-based antenna patterns of the detector. Though it is not discussed here, one must assume the wavelength of the gravitational wave is much larger than the detector in order to ignore the frequency in the antenna patterns, GW see Refs. [22,24] for more information. If the signal’s natural x-axis, eˆx , is instead GW rotated an angle ψ relative to xˆ about eˆz , then you can rotate the polarization 0 0 tensors and determine that the new polarization tensors, eˆ+ and eˆ×, are

0 eˆ+ = cos(2ψ)ê+ − sin(2ψ)ê×, (1.98) 0 eˆ× = sin(2ψ)ê+ + cos(2ψ)ê×. (1.99)

We can apply the same transformation to the antenna patterns derived above to obtain

1 F (θ, φ, ψ) = (1 + cos2 θ) cos(2φ) cos(2ψ) − cos θ sin(2φ) sin(2ψ), (1.100) + 2 1 F (θ, φ, ψ) = (1 + cos2 θ) cos(2φ) sin(2ψ) + cos θ sin(2φ) cos(2ψ). (1.101) × 2

1.5 Introduction to Compact Binary Data Analysis Methods

Compact binary data analysis relies on a two-hypothesis model. We assume that in the absence of a gravitational wave signal, the data measured are sampled from an underlying probability distribution, which we refer to as the noise distribution. If the signal we are searching for is n points long, our two possible hypotheses are that each consecutive set of n data points are entirely sampled from the noise distribution, or they are the sum of a gravitational wave signal and points sampled from the noise distribution. The former hypothesis is the noise hypothesis and is denoted by N , the latter is the signal hypothesis and is denoted by S. Given a set of n datapoints, ~x = {xi | i = 0, 1, . . . , n − 1}, we seek the conditional probability of the signal hypothesis. This is called the posterior probability and is denoted by P(S| ~x). We can use Bayes’ theorem to express the signal posterior probability in terms of the signal likelihood function P(~x| S), the probability of observing the datapoints ~x given the signal hypothesis; the signal prior P(S), the a priori probability

27 of the signal hypothesis being true; and the evidence P(~x), a normalization term, which for the two-hypothesis model is P(~x) = P (~x| S)P(S) + P (~x| N )P(N ),

P(~x| S)P(S) P(S| ~x) = . (1.102) P(~x)

Computing the signal likelihood function turns out to be computationally expensive for the case of compact binary signals. This is because we have no way of knowing, a priori, what signal we will observe before we observe it. As a consequence of this, we must search for a family of signals instead of a single signal, which can be accomplished by exploring the signal parameter space via Monte Carlo methods. Once the signal likelihood function is computed, converting to the posterior is as easy as picking an appropriate prior distribution and normalizing the product of the prior and the likelihood. Early gravitational-wave data analysts realized that contemporary computers were not powerful enough to look for signals using this method, so they created a two-tier system for finding compact binary signals: identification and parameter estimation. The LIGO detectors are noise dominated; even at design sensitivity it is expected that the overwhelming majority of measurements will be noise for compact binary analyses. Because of this, identification pipelines search for data that look statistically distinct from the rest of the data (which we assume to be noise), and then parameter estimation pipelines explore the full compact binary parameter space around times identified by the identification pipeline. This chapter will only cover identification pipelines. Although we cannot compute the evidence in advance, we can use Bayes’ theorem to express it in terms of the noise posterior probability, the noise likelihood function, and the noise prior. Rearranging, we get

P(S| ~x) P(~x| S) P(S) = . (1.103) P(N | ~x) P(~x| N ) P(N )

From this we see that, for a given set of data ~x, the ratio of the signal to noise posteriors is proportional to the likelihood-ratio,

P(~x| S) L (~x) ≡ . (1.104) P(~x| N )

The likelihood-ratio is often used when analysts want to know how likely the noise

28 is to produce a given observation, which is exactly what identiﬁcation pipelines strive to do. In the following sections, we will derive a term that maximises the likelihood-ratio for the standard, normal case, and then for an arbitrary Gaussian case. In both cases, we assume the noise is stationary, meaning that the underlying probability distribution is unchanging over time, or that the moments (the mean and the variance, for example) are not changing.

1.5.1 Matched Filtering

1.5.1.1 Standard Normal Noise Distribution

In the case of uncorrelated Gaussian noise with a mean zero and variance 1, the likelihood of measuring the points ~x = {xi, | i = 0, 1, . . . , n − 1} is

"N−1 −x2 # P(~x| N ) = (2π)−N/2 exp X i . (1.105) i=0 2

Under the signal hypothesis, the data are a sum of samples from the noise distribu- ~ tion and a gravitational-wave signal h = {hi | i = 0, . . . , n − 1}. The model of the signal we are looking for, ~h, is referred to as a template. Assuming you know the exact signal that is in the data, the diﬀerence of the data and the signal will follow the noise distribution, which tells us that the likelihood of measuring ~x assuming the signal hypothesis is

"N−1 −(x − h )2 # P(~x| S) = (2π)−N/2 exp X i i (1.106) i=0 2

Calculating the likelihood-ratio, Eq. 1.104, we get

"n−1 # X 1 2 L (~x) = exp xihi − hi . (1.107) i=0 2

For a given signal, h(t), the likelihood-ratio is maximized by the dot product Pn−1 i=0 xihi. We divide this quantity by its RMS value under the noise hypothesis to deﬁne a signal-to-noise ratio, ρ. The RMS value under the noise hypothesis can

29 be computed analytically for stationary, Gaussian noise,

v v u* !2+ u u n−1 un−1 n−1 u X uX 2 2 X X t nihi = t hni ihi + 2 hninjihihj (1.108a) i=0 i=0 i=0 j6=i v un−1 uX 2 , = t hi , (1.108b) i=0 where the average is taken over many iterations of the noise. By convention, we Pn−1 2 normalize the signal so that i=0 hi = 1, thus

n−1 X ~ ρ = hixi = h · ~x. (1.109) i=0

The SNR, ρ, maximizes the likelihood-ratio, which maximises ratio of the signal posterior to the noise posterior, conceptually this means that the higher the SNR, the more likely it is that there is a signal h in the data. If our dataset has a longer duration than the signal we are searching for, as is often the case, we can deﬁne the time series ρ(t) as the cross-correlation between the data and the signal. The discrete SNR time-series is then

n−1 X ρj = xi+jhi, j = 0, 1,...,L − n, (1.110) i=0 where L is the number of sample points in ~x, i.e. ~x = {xi | i = 0,...,L − 1}.

1.5.1.2 Normal Noise Distribution

Eq. 1.110 assumes the noise is a stationary, uncorrelated, Gaussian process with null mean and unity variance, but there is no reason to assume that the advanced LIGO noise distribution is uncorrelated or has unity variance, though we will assume for now that the noise is stationary. Thus we need to modify our noise distribution, using the noise temporal covariance matrix Σij = hninji instead of 2 2 only its variance σi = Σii = hni i,

 n−1 n−1  n −1/2 1 X X xixj P(~x| N ) = ((2π) det (Σ)) exp −  . (1.111) 2 i=0 j=0 Σij

30 Propagating this change through the same steps as before, we see that the discrete SNR time series becomes

n−1 n−1 X X xi+jhk ρj = , j = 0, 1,...,L − n (1.112) i=0 k=0 Σi+j,k where the comma in the index of the covariance matrix is to make reading the indices easier, and the normalization condition for the template, h, is now deﬁned by reference to the covariance matrix. Fortunately, the Wiener-Khintchin theorem [24], allows us to simplify Eq. 1.112. For stationary processes, the covariance only depends on the amount of time between samples, not on the samples themselves; i.e. the covariance matrix, Σij, only depend on |j − i|. In the continuous limit, the Wiener-Khintchin theorem tells us the autocorrelation of a stationary process is one-half the Fourier transform of the single sided PSD, Sn(f),

1 Z ∞ hn(t)n(t + τ)i = df Sn(|f|) exp [−2πifτ] (1.113) 2 −∞ where Sn(f) is deﬁned such that

1 hn˜(f)n ˜∗(f 0)i = S (f) δ(f − f 0), f > 0, (1.114) 2 n where the average, h· · · i, is taken over many iterations of the noise, and

Z ∞ n˜(f) = dt n(t) exp [2πitf] (1.115) −∞ is the Fourier transform of the noise distribution [24]. Conceptually, the PSD is the average square-magnitude of the noise distribution as a function of frequency. We will assume throughout this dissertation that the noise distribution is ergodic, meaning that its temporal average is equal to the ensemble average (an average taken over many iterations of the noise). The ergodicity assumption allows us to measure the PSD, even though we only measure one iteration of the noise. The PSD measurement is discussed more in Ref. [4]. Eqs. 1.113 and 1.114 tell us that stationary processes are uncorrelated in the frequency domain. This means, under the noise hypothesis, we can remove the temporal correlation of the noise in the frequency domain. This process is called

31 whitening, because the result is white noise (noise which does not have any preferred frequency). Although Eqs. 1.113 and 1.114 are written in the continuous limit, we will work through the whitening process in the discrete limit, since real data is always discrete. We will ﬁrst outline the whitening process, and then we will show that whitened data is uncorrelated with a variance of 1, like our example case (Sec. 1.5.1.1. The whitening process is as follows:

1. Fourier transform the time series data.

2. Divide the Fourier transform by the square root of one half of the single-sided PSD.

3. Inverse Fourier transform the frequency series.

To show that this works, ﬁrst we must deﬁne the discrete Fourier transform and its inverse, taking care to ensure each transformation is unitary (i.e. it preserves the inner product),

n−1 1 X x˜k = √ xj exp [2πijk/n] , (1.116a) n j=0 n−1 1 X xj = √ x˜k exp [−2πijk/n] . (1.116b) n k=0

Mathematically, the whitening transformation is

1 n−1 n−1 x exp [2πimk/n] x → xˆ = X X m exp [−2πkj/n] , (1.117) j j n q k=0 m=0 Sk/2 where aˆ is the notation that will be used in this thesis to denote a whitened time series, and the PSD is divided by two because it is the single sided PSD (i.e. it is normalized using its integral over positive frequencies, so when summing over all of the frequencies instead of just positive we must divide it by 2). Now we can ∗ compute the correlation, hxˆj xˆj+ki in terms of the Fourier transformed variables,

n−1 n−1 ∗ ∗ 2 X X hx˜` x˜mi hxˆj xˆj+ki = √ exp [2πi`j/n] exp [−2πim(j + k)/n] (1.118a) n `=0 m=0 S`Sm 2 n−1 n−1 1 S δ = X X √2 ` `m exp [2πi`j/n] exp [−2πim(j + k)/n] (1.118b) n `=0 m=0 S`Sm

32 1 n−1 = X exp [−2π`k/n] (1.118c) n `=0

= δk0, (1.118d) where in going from the ﬁrst line to the second line we used Eq. 1.114 in the discrete limit. An example of stationary, Gaussian LIGO data before and after whitening is shown in Fig. 1.2. We can see now that the whitening process transforms the data

Figure 1.2. An example of the whitening process. The plot on the left shows 8 seconds of stationary, Gaussian noise that is not white. The plot on the right shows the same data after it has been whitened. The data used to create this plot were downloaded from the Gravitational Wave Open Science Center [2]. such that it is uncorrelated and has a variance of 1, which allows us to make use of Eq. 1.110 once again, although hj also needs to be transformed using Eq. 1.117 because a signal in the data will be altered when the data are whitened. We can now re-write Eq. 1.110 in terms of the PSD and Fourier transformed variables,

n−1 ˜∗ X hkx˜k ρj = 2 exp [−2πikj/n] . (1.119) k=0 Sk

In the literature, Eq. 1.119 is often written in the continuous regime in one of the following ways,

Z ∞ h˜∗(f)x ˜(f) ρ(t) = 2 df exp [−2πift] , (1.120a) −∞ S(|f|) Z ∞ h˜∗(f)x ˜(f) = 4 Re df exp [−2πift] , (1.120b) 0 S(f)

33 Z ∞ = dτ hˆ(τ)x ˆ(t + τ), (1.120c) −∞ where aˆ is a whitened time series as described above. The t = 0 case in the frequency domain is often used as a deﬁnition for the inner product in compact binary data analysis,

Z ∞ a˜∗(f)˜b(f) ha | bi ≡ 2 df , (1.121a) −∞ S(|f|) Z ∞ a˜∗(f)˜b(f) = 4 Re df , (1.121b) 0 S(f) Z ∞ = dτ aˆ(τ)ˆb(τ) (1.121c) −∞

In the discrete limit, this inner product becomes

n−1 a˜∗˜b ha | bi = 2 X k k , (1.122a) k=0 Sk n−1 X ˆ = aˆjbj. (1.122b) j=0

There are two important observations to note about Eqs. 1.119 and 1.120. Firstly, they can be described as frequency-weighted convolutions of the data with a template, performing the inner product such that frequency bins that are less noisy (where the PSD is less) count for more than noisier frequency bins. Secondly, ρ(t) h˜∗(f)˜x(f) is the inverse Fourier transform of 2 S(f) , which means a naive matched-filtering algorithm can be performed efficiently in the frequency domain with use of FFT algorithms. In addition to not knowing the time of the samples which contain the signal, for compact binary signals there is also an unknown phase, φ. To measure this phase, instead of matched-filtering a single template, we use two templates: a cosine-like template (often referred to as the in-phase component) and a sine-like template (the quadrature-phase component). We use the output of the former as the real part of a complex SNR time-series, z(t), and the latter as the imaginary part,

n−1 X ˆc ˆs zj = xˆm+j hm + ihm , j = 0, 1,...,L − n, (1.123) m=0

34 where hc(t) is the cosine-like template and hs(t) is the sine-line template. The phase is measured by taking the argument of the complex SNR, while the SNR of the template at that phase is the amplitude of the complex SNR,

Im(zj) φj = arctan , (1.124a) Re(zj) q 2 2 ρj = Re(zj) + Im(zj) . (1.124b)

1.5.1.3 Real LIGO Data

In the case of stationary, Gaussian noise, the matched-filter is the optimal filter in the sense that higher output SNR always means a higher probability of the data containing the waveform. However, real LIGO data contain many periods of non-stationary data, typically called glitches. One example of a glitch is shown in Fig. 1.3. The glitch shown in Fig. 1.3 has a max strain amplitude of about 759. Whitened data should have a mean of zero and unity variance, thus this glitch is a 759σ excursion from stationary, Gaussian data. Additionally, whitened data should be uncorrelated from time-sample to time-sample, which is clearly not true for the glitch shown in Fig. 1.3. Excursions such as this are extremely common, occurring several times per hour if not more often. The consequence of having so many glitches in the data is that matched-filtering alone is not sufficient to identify quiet signals in the data, as glitches can produce SNRs of 100 or higher. Different search pipelines handle glitches in different ways, the methods used by the GstLAL-based inspiral pipeline will be discussed more in Sec. 1.6. Additionally, the LIGO data constrains the gravitational-wave frequencies we can search for. Initial LIGO was sensitive to frequencies down to 40 Hz, while advanced LIGO at design sensitivity will be sensitive to frequencies down to 10 Hz. At the time of writing, the detectors have some sensitivity down to 15 Hz, though its most sensitive frequencies are those from ∼ 100 Hz to ∼ 300 Hz. A PSD measured from public data taken December 22, 2015 is shown in Fig. 1.4, though it should be noted that the detectors only had sensitivities down to ∼ 20 Hz at the time.

35 Figure 1.3. A noise transient, commonly referred to as a glitch, from December 22, 2015. Whitened data should have a mean of zero and a variance of one, thus the peak amplitude of this glitch is a several-hundred σ excursion from stationary, Gaussian data. Whitened data should also be uncorrelated from one time sample to the next, which is clearly not true near the glitch. Glitches such as this occur several times an hour in real LIGO data. The data used to create this plot were downloaded from the Gravitational Wave Open Science Center [2].

1.5.2 The p-value

In the previous section, we showed that the matched-ﬁlter output, ρj, maximises the signal posterior probability in the case of stationary, Gaussian noise. In other words, the higher ρj is, the more likely it is xj+i is the sum of hi and a sample drawn from the noise distribution. Under these conditions, ρj can be used a ranking statistic; a ranking statistic is a test-statistic that monotonically increases with the signal posterior. This means that if you measure two triggers with SNRs

ρa and ρb, you can conﬁdently say which is more likely to contain the signal. Here a trigger is a single measurement of the SNR, in other words a trigger is a list of 3 values: the time of the trigger, the phase, and the SNR. We are assuming we do not have any information about the signal model, but we do have information about our noise model. If we cannot get the posterior probability of the signal

36 Figure 1.4. The Power Spectral Density (PSD) measured from data taken from the morning of December 22, 2015, at the Hanford, WA detector. The PSD represents the average square-magnitude of the noise as a function of frequency, and is used to weight the matched-ﬁlter such that frequency bins where the noise is lower are more heavily weighted than bins where the noise is higher. The data used to create this plot were downloaded from the Gravitational Wave Open Science Center [2]. hypothesis for a given set of data, the next best thing is to understand how likely we are to measure a given SNR under the noise hypothesis. To measure this, we use the p-value, which is deﬁned for a given test-statistic, noise hypothesis, and experiment. The p-value of a trigger from an experiment is the probability of measuring a test-statistic at least as high as the trigger’s test-statistic assuming that all of the triggers from the experiment are drawn from the noise distribution. In this section, we will work out the p-value for the stationary, Gaussian case, as a function of SNR. As we have shown above, once the data have been whitened, the signal free portions of data are temporally uncorrelated with a mean of 0 and unity variance. Under the Gaussian assumption, the real and imaginary parts of the complex-SNR are also Gaussian distributed with a mean of 0 and unity variance, thus ρ2 follows a χ2-distribution with 2 degrees of freedom,

2 − ρ P(ρ| N ) dρ = ρe 2 dρ, ρ ∈ [0, ∞). (1.125)

q π 2 under the noise hypothesis, hρi = 2 and hρ i = 2. P(ρ| N ) is a probability

37 density, meaning that the probability of measuring an SNR of ρ is zero, you can only compute non-zero probability for an interval of SNRs. We thus want to use the complementary cumulative distribution function (CCDF). The CCDF is the probability a measuring a random variable ρ∗ that is at least as large as ρ when sampling Eq. 1.125

Z ∞ P(ρ∗ ≥ ρ| N ) ≡ dρ0 P(ρ0| N ) = e−ρ2/2. (1.126) ρ

Eq. 1.126 gives us the probability of measuring an SNR at least as large as ρ if we only measure one SNR, but if we do not know where the signal is, we need to ask what the probability of measuring an SNR at least as large as ρ after sampling the noise distribution (Eq. 1.126) M times, where M is often referred to as the trials factor. For example, the probability of measuring an SNR at least as high as 4 under the stationary-Gaussian noise hypothesis is P(ρ∗ ≥ 4| N ) ≈ 3.4 × 10−5, but what if we sample Eq. 1.126 at a rate of 2048 Hz for a second? To model this, we ﬁrst assume that this process is Poissonian and assume that each sample is independent of the previous under the noise hypothesis. The rate, in units of experiment−1, at which we expect to draw events with an SNR ≥ ρ after drawing M samples is MP(ρ∗ ≥ ρ| N ), thus the probability of drawing at least one sample point with an SNR ≥ ρ after drawing M samples from the noise distribution is unity minus the probability of not measuring any samples with an SNR ≥ ρ,

∗ ∗ P(ρ ≥ ρ| N1,..., NM ) = 1 − exp [−MP(ρ ≥ ρ| N )] , (1.127)

where Ni is the noise hypothesis for the trigger associated with ρi. Thus, after drawing samples from the noise distribution 2048 times the probability of drawing ∗ at least one sample with an SNR greater than 4 is P(ρ ≥ 4| N1,..., NM ) ≈ 0.50 8. This is the p-value of a trigger with an SNR of 4 from our stationary, Gaus- sian example experiment. The p-value tells us that although measuring an SNR of 4 or higher seems rare, it turns out to be relatively common after not many measurements. 8Note that this is assuming we are drawing independent samples from a Gaussian distribution of SNR. If the SNR samples are actually computed via matched-ﬁltering, then samples near each other in time will be correlated through their use of the same template and the computed p-value can be thought of as an upper limit.

38 The p-value is unitless, but it can be considered to have units of “per experiment”. For example, the p-value of 0.50 that was derived above can be interpreted as “One expects 0.50 triggers with an SNR of 4 or higher each time they sample the SNR distribution 2048 times.” It is important to remember that the p-value is defined for a given test statistic, experiment, and noise hypothesis. The p-value of measuring an SNR of 4 or greater will be higher, for example, if your experiment contains more samples. Defining the p-value on a per experiment basis accounts for the physical truth that sampling a distribution more often leads to a higher probability to sample the tail of the distribution; the more samples we have, the more likely it is we will see something “loud” (rare, as defined by the test statistic) from the noise.

1.5.3 Template Banks

As discussed in the introduction to Sec. 1.5, we do not know what compact binary signal we will detect until we detect it. Instead of searching for one signal, we must search for a family of signals. To choose which signals we want to search for, we must choose a region of the signal parameter space that we are interested in. For example, we could decide we only care about signals from non-spinning compact binaries with component masses between 1 M and 2 M . Gravitational-waves from compact binary mergers depend strongly on the masses of the components, thus there is still an infinite number of different templates that could be chosen from this region of the parameter space. Before proceeding, we must define a few terms. A template bank is a collection of templates used to search for gravitational-wave signals. Template banks span a region of the signal parameter space and are discrete, in that they contain templates that represent discrete points in the continuous parameter space. The points chosen from the parameter space for inclusion in the template bank are chosen such that the matched-filter inner-product (Eq. 1.122) of a template from any point in the parameter space with a template representing one of the discrete points will be at least some value δρ. δρ is the minimal match, and mathematically

δρ(h) = maxhh(t) | hii, (1.128) i where h(t) is an arbitrary template from the region of parameter space we’re

39 interested in and the index i enumerates the templates in the template bank. The minimal match sets an upper limit on the amount of SNR we are willing to lose by not having exactly the right waveform. It is currently set to 0.97, which means we should measure at least 97% of the ‘true’ SNR for any signal in the parameter space spanned by our template bank. An example template bank is shown in Fig. 1.5. Template banks, their discretization, and their generation are

Figure 1.5. A plot of the component masses of a template bank. Every point represents a diﬀerent template used for matched-ﬁltering. More details can be found on this template bank in Sec. 4.2.1. This plot was originally published in Ref. [3]. discussed in more detail in Refs. [44,45].

1.5.4 Horizon Distance, Eﬀective Distance, and Range

It is often helpful to discuss the sensitivity of the detectors in terms of the maximum detection distance for a given signal. In order to do this, one must first choose what source parameters to use for the signal. One convention is to use a canonical binary neutron star system, where both components are 1.4 M , so often if the literature does not specify a system it is using this one. There are two metrics commonly used to measure how far away a detector can observe a signal from: the horizon distance and the range. The horizon distance is most easily defined using a quantity called the effective distance. The effective distance, Deff , is the distance that would be measured from a compact binary merger gravitational-wave signal if the binary were optimally oriented, i.e. if the orbital plane were parallel to the

40 detector plane. From Ref. [46],

−1/2  2 !2  2 1 + cos ι 2 2 Deﬀ = D F + F cos ι , (1.129) + 2 × where D is the physical distance. The horizon distance is the furthest away a detector should expect to observe a signal with an SNR of 8, thus [46]

ρ D D = 0 eﬀ , (1.130) H 8 where ρ0 is the nominal SNR, the SNR in the absence of noise (i.e. if the matched- ﬁlter were just the signal and the normalized template). The horizon distance can be estimated using a PSD and a template for the signal. The range is then the average distance a detector expects to observe a signal with SNR 8. Analytically, the range can be approximated by dividing the horizon distance by 2.26 [22], though it is also often estimated by inserting fake signals into data and measuring the detected fraction as a function of distance [4].

1.6 The GstLAL-based Inspiral Pipeline

The first compact binary search pipelines written by members of the LIGO Scien- tific Collaboration performed matched-filtering in the frequency domain because it is computationally cheaper to compute the SNR as the inverse Fourier-transform of the inner product in the frequency domain using Fast Fourier Transform (FFT) algorithms than it is to naively compute the convolution in the time domain (see Eq. 1.120a and 1.120c). However, frequency-domain calculations are inherently high-latency. Low latencies can be achieved at a computational cost, however matched-filtering in the time domain is much cheaper to perform in low latency. The GstLAL-based inspiral pipeline performs matched-filtering in the time domain via the LLOID algorithm.

1.6.1 The LLOID Algorithm

The Low-Latency Online Inspiral Detection (LLOID) algorithm is the name of the matched-ﬁltering algorithm used by the GstLAL-based inspiral pipeline. It is

41 discussed in detail in Refs. [4, 47], and will only be briefly summarized here. The LLOID algorithm is designed to lower the computational cost of matched-filtering in the time-domain to match or beat the cost of performing matched-filtering in the frequency domain with FFT algorithms. This is accomplished by use of critical sampling and linear algebra methodology.

1.6.1.1 Critical Sampling

The ﬁrst cost-saving measure requires an understanding of the Nyquist sampling theorem, which states that if you want to fully resolve a signal at a given frequency f, you must sample it at a sample rate of at least 2f. A naive search for the inspiral portion of a compact binary gravitational-wave signal would need to be sampled at a rate of 2fISCO or higher. Modern compact-binary search pipelines analyze data sampled at 2048 Hz, meaning that they only search for signals up to frequencies of 1024 Hz, a decision made to focus on the most sensitive region of the LIGO noise curve. This is not enough of a savings to make matched-ﬁltering in the time-domain worthwhile, as 2048 operations per second still need to be performed for the duration of the signal to produce even one measurement of the SNR, and compact binary signals can be up to minutes long in the LIGO frequency band. Fortunately, the entire signal does not need to be sampled at this rate. The frequency of compact binary gravitational-wave signals increases monotonically with time, so the beginning of the template can be sampled at a lower frequency than the end of the template. The GstLAL-based inspiral pipeline uses this to slice the template up into several components, each sampled at twice the highest frequency in the slice rounded up to the nearest power of 2. The data are then down-sampled, a process by which the 2048Hz time-series is converted to a time series with a lower sample rate, to match the sample rates of the sliced templates. Each component of the complex SNR, at a single point in time, is an inner product and can be rewritten as a sum of inner-products over the template slices, nS −1 X X ˆ ha | bi = aˆSjbSj, (1.131) S j=0

th th where aˆSj is the j sample in in the S slice, which has sample frequency fS and

42 nS sample points, where X n = nS (1.132) S is the total number of samples in the original template. To compute the SNR time series, as opposed to a single SNR, we break the convolution up into diﬀerent frequencies, so that we have as many SNR time-series as we have slices of template. We then up-sample the time-series, a process by which a time-series is transformed such that its sample rate increases, and add them together such that we recover ρ(t) sampled at the original rate. For more details on this procedure, see Refs. [4,47].

1.6.1.2 Singular Value Decomposition

Another source of large computational cost in matched-ﬁltering is the number of templates that are searched for. Repeating the convolution for every template is expensive when dealing with thousands of templates, not to mention the hundreds of thousands of templates used in advanced LIGO searches. Fortunately, it turns out that collections of similar templates, where similarity can be measured in diﬀerent ways, generally form an overcomplete basis [48, 49]. In this case, the full collection of N templates can be represented by N 0 < N basis vectors, which can be found via a matrix decomposition known as the singular value decomposition [4, 47,49]. The SVD decomposes a matrix into a reconstruction matrix, a set of scalars known as singular values, and a set of orthogonal basis vectors. There is one singular value for each orthogonal basis vector, and the singular value is directly proportional to the importance of its associated basis vector in the reconstruction of the original matrix. If the original matrix is overcomplete, meaning its columns and rows are linearly dependent sets of vectors, then the set of basis vectors returned by the SVD will be smaller than the set of linearly dependent vectors that make up the matrix. For our waveform templates, this means we can represent N templates with N 0 < N orthogonal basis vectors in the SVD basis. Furthermore, we can use the singular values to discard selected orthogonal basis vectors that are least important to the reconstruction. The number of orthogonal basis vectors to discard can be experimentally found by requiring that the inner product of the original

43 template with the reconstructed template is

hhorig | hreconsi ≥ 1 − , (1.133) where is a user-defined tolerance. Remember that the template is normalized such that the inner product of it with itself is 1, thus the tolerance sets the ratio of SNR lost in the reconstruction. The SVD allows us to decompose the template bank into a smaller number of templates that we can use for matched-filtering. Now instead of convolving the data with N templates, we can convolve the data with N 0 < N templates in a different basis and then use the reconstruction matrix to obtain the convolutions in our original basis.

1.6.1.3 The LLOID Template Decomposition

The LLOID algorithm’s decomposition of templates is show in Fig. 1.6. The templates are ﬁrst sliced in time and down-sampled, as described in Sec. 1.6.1.1. The templates are zero-padded such that they are all same length (as each template may have a slightly diﬀerent duration in our frequency band), and the partitioning of the sliced templates is the same across the entire collection, i.e. if the highest sample-rate slice has n0 samples in one template, the highest sample-rate slice has n0 samples in every template in the group we are decomposing. Then each collection of slices are decomposed via the SVD, i.e. all of the highest sample-rate slices are combined to make a matrix that is then decomposed via the SVD. This process is described graphically in Fig. 1.6.

1.6.2 Signal-Consistency Test Parameter ξ2

Although we showed earlier that the matched-ﬁlter SNR is an optimal test-statistic for stationary, Gaussian data, we can still improve our detection ability in non- stationary, non-Gaussian noise by evaluating another test statistic. The test- statistic used by the GstLAL-based inspiral pipeline uses an autocorrelation-based test statistic referred to as ξ2. Conceptually, the idea is that if your template matches the signal in the data, then the convolution of the template with the data that contains the signal should look like the autocorrelation of the template with

44 Figure 1.6. The template decomposition used by the LLOID algorithm. The top of the figure shows three inspiral templates, with sections color-coded to represent the time- slices, which are sampled at different rates. The bottom of the figure shows examples of the orthogonal basis vectors returned by the SVD of the sliced templates. The templates in the SVD basis are convolved with the data in the matched-filtering process, though the final result of the LLOID algorithm is the SNR time-series in the original basis, sampled at the sample rate of the data. This figure originally appeared in Ref. [4]. some additive noise. For this next section, we will assume that the phase φ (Eq. 1.124a) is known to be φ0 and choose the time of the signal to be t = 0 by convention. If we rotate our complex SNR time-series by −φ,

z(t) → z0(t) = z(t)e−iφ0 (1.134) then z0(0) = ρ, where ρ is given by Eq. 1.124b. Under the signal hypothesis, the SNR time-series around this point should look like the autocorrelation function,

Z ∞ |h(f)|2 R(t) = 2 df e−2πitf . (1.135a) −∞ Sn(|f|)

Notice that nothing is said about phase here, the autocorrelation of in-phase and quadrature-phase components of a signal (the cosine-like and sine-like terms) are

45 identical. If there were no noise, z0(t) = z0(0)R(t), therefore a quantity of interest to us is ξ2(t) = |z0(t) − z0(0)R(t)|2. (1.136)

At this point, we will plug the original SNR time-series back in and see that ξ2(t) is conserved under transformations of the type described by Eq. 1.134,

ξ2(t) = |z(t)e−iφ0 − z(0)e−iφ0 R(t)|2, (1.137a) = |z(t) − z(0)R(t)|2. (1.137b)

ξ2(0) = 0 by design, but the interval around t = 0 can tell us how close the output of the matched-ﬁlter is to the autocorrelation of the template. To turn this time- series into a scalar test-statistic, we integrate it over a short time-interval around t = 0 and normalize it by the expectation value in noise [4],

P2δ |z − z R |2 ξ2 = j=0 j δ j , (1.138) P2δ 2 j=0(2 − 2Rj ) where δ is a number experimentally found; currently we use δ = 351 with a 2048 Hz sample rate. Notice that the t = 0 term is the j = δ sample.

1.6.3 Trigger Generation

Once the SNR time-series are computed, the highest peak from each 1 second window for each template in each detector is saved as a “trigger” if it passes an SNR trigger threshold, typically set to 4. Initially the trigger consists of the template used for the trigger, the time, and the signal-consistency test statistic, ξ2. Only triggers that were coincident in time and template across multiple detectors were considered as candidates until the middle of O2. In the low-latency analysis, the SNR time-series immediately surrounding each trigger in a given detector is saved until it is determined whether the candidate will be uploaded to an online database [7], which will require the SNR time-series to use for localization, after which it is discarded by the analysis.

46 1.6.4 GstLAL’s Likelihood-Ratio

Sec. 1.5.2 assumed the noise distribution is stationary and Gaussian when computing the p-value, however it has been experimentally shown that this is rarely a good assumption for the LIGO noise distribution. While LIGO’s noise contains a Gaussian component, is not actually Gaussian; it also contains many small periods of non-stationarity. Data quality experts attempt to identify intervals of non-stationarity that can be clearly shown to come from couplings inside the detector, which are then excluded from analysis by the search pipelines. However, they cannot identify all periods of non-stationarity; the methodology to account for the non-Gaussian, non-stationary behavior when searching for gravitational- wave signals is what currently distinguishes compact binary search pipelines from one another. The GstLAL-based inspiral pipeline uses an internal veto-system to excise data which appears to not be stationary, and uses the likelihood-ratio (Eq. 1.104) as a ranking statistic, where the noise model is informed by the data instead of assumed to be Gaussian. The pipeline’s implementation of the likelihood-ratio does not use the data, ~x, directly; the data are first matched-filtered, and the output of the matched-filter informs the ranking statistic. In other words, the likelihood-ratio used by the pipeline is a function of SNR, not a function of ~x like when it was introduced. The pipeline’s likelihood-ratio is a function of many parameters, including the signal- consistency test parameter, ξ2 (Sec. 1.6.2), and the sensitivity of the detectors at the time of a candidate. In principle, any piece of information available to the analysis can be used as a parameter in the likelihood-ratio, it only needs to know how that information is distributed under the noise and signal hypotheses. For example, the likelihood-ratio used to rank candidates at the time of writing incorporates the SNRs, the signal-consistency test parameters, ξ2 (Sec. 1.6.2), the sensitivity of the detectors that observed the candidate, the time the candidate was observed, the set of detectors that observed the candidate (e.g. was it seen by 2 detectors or 1 detector), the difference in arrival times and phases across the detectors (if more than one detector observes it), and the template that found it. In order to compute the likelihood-ratio, it is factored into a sum of many distributions using the probability chain rule,

P(a, b| c) = P (a| b, c)P(b| c) . (1.139)

47 In this section, we will only factor the likelihood-ratio used in the initial detection of gravitational waves (Ch. 2). We will discuss new terms only when necessary to discuss a detection. At the time of the GW150914 detection, the likelihood-ratio was ~2 ~ ~ ¯ ¯ P ~ρ, ξ , DH, O θ, S P θ S ~2 ~ ~ ¯ L ~ρ, ξ , DH, O, θ = , (1.140) ~2 ~ ~ ¯ ¯ P ~ρ, ξ , DH, O θ, N P θ N where the coordinates of the vectors are the detectors involved in the candidate ~ being ranked, DH are the horizon distances of the detectors that observed the candidate, and O~ are the detectors that observed the candidate, and θ¯ is an index that denotes the template that found the candidate. θ¯ has been factored out to call special attention to it, as we will not carry the parameter through out calculations ¯ because we will mostly ignore it until Ch. 8. Until then, assume P θ S and ¯ P θ N are uniform until explicitly said otherwise. The numerator of the likelihood-ratio used in the initial detection of gravitational waves was factored as

~2 ~ ~ ~ ~ ~ P ~ρ, ξ , DH, O S = P DH P O DH, S ~ ~ Y 2 × P(~ρ| DH, O, S P ξIFO ρIFO, S . (1.141) IFO∈{H1,L1} while the denominator was factored as

~2 ~ ~ ~ ~ Y 2 P ~ρ, ξ , DH, O N = P DH P O N P ρIFO, ξIFO N . (1.142) IFO∈{H1,L1}

~ Notice that the horizon distance term, P DH , is common between the numerator and denominator, and thus does not aﬀect the likelihood-ratio. It is necessary, however, if one wishes to sample the signal or noise model by sampling the numerator and denominator respectively. Many of the distributions in Eq. 1.141 and 1.142 are computed analytically or semi-analytically, as described in Ref. [46], however the one term that is not is the SNR, ξ2 term in the denominator. In order to estimate this, we separate the templates into diﬀerent bins; the binning was done

χ1m1+χ2m2 on the eﬀective spin, χ = where χ1 and χ2 are the dimensionless com- m1+m2 ponent spins, and chirp mass, M in the analysis this version of the likelihood-ratio was used in. We assume that all of the templates in a given bin have the same

48 2 response to noise, and estimate P(ρIFO, ξIFO| N ) by histogramming non-coincident triggers, then smoothing and normalizing the histogram [46]. Thus the SNR, ξ2 term in the denominator is the term that captures the non-Gaussianity of the noise for each detector. A background is estimated using Gaussian statistics before any information is collected, but is weighted such that the counts in the histogram will dominate the normalization. The purpose of this is so that if a coincident candidate arises with (ρ, ξ2) values where the noise histogram doesn’t have any counts, the likelihood-ratio does not produce a divide-by-zero inﬁnity.

1.6.5 Estimating Signiﬁcance

To compute the p-value, we must ﬁrst compute the distribution of our ranking statistic under the noise hypothesis. In our Gaussian noise example in Sec. 1.5.2, we used SNR as a ranking statistic and computed the p-value analytically. The GstLAL-based inspiral pipeline cannot do this, not only because its likelihood-ratio cannot be written down in an analytical form, but also because the noise model (the denominator of the likelihood-ratio) is measured from the data. We compute P (log L∗ ≥ log L| θ,¯ N for each template bin by sampling the noise model (the denominator of the likelihood-ratio), assigning the sample a likelihood-ratio, and then histogramming it. We use an importance-weighted sampling method to ensure that our sampler probes the tails of the log L noise distribution. We then marginalize over all of the bins,

Z ∗ ¯0 ∗ ¯0 ¯0 P (log L ≥ log L| N ) = dθ P (log L ≥ log L| θ , N P θ N , (1.143) θ¯

¯0 where P θ N is estimated by the fraction of the number of candidates in that bin compared to the total number of candidates observed across all bins. The index θ¯ here is most easily thought of as an index for the template bin, templates were not treated diﬀerently by the likelihood-ratio other than the bins they were in for this analysis. The marginalized distribution will describe the distribution of likelihood- ratios under the noise hypothesis, however it assumes that every sample you’ve drawn from that distribution is independent, which is not the case as the templates are correlated both in time and with each other. To produce a list of independently drawn candidates, we only keep the candidates with the highest likelihood-ratio in any given 4-second window, a procedure we call clustering. However, the Monte

49 Carlo sampler used to produce the bin-dependent noise distributions of log L does not simulate this part of the analysis. This manifests itself as the sampler not being able to properly model the low-log L regime, where candidates are so likely to be close in time to a louder candidate (as measured by the likelihood-ratio) that not enough are recovered to clearly resolve the distribution; this eﬀect is referred to as extinction. The low-likelihood-ratio extinction eﬀect can be seen in Fig. 1.7.

Figure 1.7. Simulated likelihood-ratios before and after clustering. Clustering removes candidates such that only the highest likelihood-ratio in any given 8 second window is preserved. This does not significantly affect the distribution of large likelihood-ratios, but it causes an extinction effect where too many low likelihood-ratio candidates are removed to reproduce the low likelihood-ratio regime without additional modeling.

There are two methods to getting around this: the first is to attempt to model this behavior as a function that takes the output of the sampler and outputs an estimate of the clustered distribution, the second is to set a likelihood-ratio threshold above the region where this effect matters (through guess and check). I attempted to model the extinction behavior as one of my first projects in graduate school, though my method was not very successful and essentially became a method to automatically compute a log L threshold to implement the second method. Once the extinction effects from clustering are dealt with, either by using an extinction model or by using a log L threshold, the observed data can be used to normalize the distribution from the sampler. Now the p-value can be computed the same way as before,

p = 1 − exp (MP (log L∗ ≥ log L| N )) , (1.144)

50 where M is ﬁnal number of observed candidates (after clustering). As we have an estimate of the rate of candidates in the analysis, MP (log L∗ ≥ log L| N ), we can also estimate the false alarm rate by dividing by the livetime, T ,

MP (log L∗ ≥ log L| N ) FAR = . (1.145) T

51 Chapter 2 | GW150914

2.1 Introduction

The first direct detection of gravitational waves almost didn’t happen. Before each observing run, the detectors participate in an exercise called an engineering run. The data from this time is analyzed by low-latency pipelines, but is not always analyzed offline. The detectors usually have a lower duty cycle because the instrument scientists are still tweaking small things. On September 14, 2015, during the last engineering run before O1, gravitational waves from the merger of a binary black hole passed through the LIGO detectors at 09:50:45 UTC. The GstLAL-based inspiral pipeline was only looking for compact binary mergers that included a neutron star at the time, due to a policy decision by the collaboration’s compact binary working group. Thankfully a burst analysis, an analysis that looks for chirp-like gravitational wave transients but doesn’t look for signals from specific sources, was running in low-latency and detected it [50]. GW150914 was identified within three minutes of passing through the detectors [51], which stopped the instrumental scientists from doing anything to change the state of the detector. The significance of GW150914 is difficult to overstate. Not only is it the first direct detection of gravitational waves, it was the first observational evidence of binary black holes, and the first observational evidence of stellar-mass black holes with masses & 25M . It also gave us the ability to experimentally estimate the rate of binary black holes for the first time, making them a much more exciting prospect for ground-based interferometric detectors. With such high uncertainties in the binary black hole merger rate, most believed that binary neutron star mergers

52 would be the first gravitational wave source to be observed. Very few expected that we would detect several binary black hole mergers before seeing even one binary neutron star merger. Even fewer expected the first detection would be a “smoking gun” detection, with well over 5σ significance.

2.2 Methods

2.2.1 Template Bank

The template bank used for offline O1 compact binary analyses was constructed using two different waveform approximates and two different placement algorithms. The bank, whose parameters are given in Table 2.1, only contained templates for binaries with component spins parallel to the binary orbital angular momentum. There are two primary reasons that the bank contains only spin-aligned templates: (1) we do not yet have waveform approximates that contain precession, and (2) including the four extra spin-dimensions of the parameter space would make existing analyses too computationally expensive to be tenable. When placing templates, templates for systems where both component masses were less than 2M were generated using the TaylorF2 approximate [38] with a minimum match of 0.97 and a geometric placement algorithm1 [6]. The rest of the bank was placed using a combination of geometric and stochastic placement2 [52]; TaylorF2 templates were used for the geometric portion of the placement, effective-one-body templates with spin-effects, referred to as SEOBNRv2 [39], were used for the stochastic portion of the placement. Note that only template placement has been discussed so far, the GstLAL-based inspiral pipeline used TaylorF2 to generate templates for systems with chirp mass less than 1.73M and SEOBNRv2 to generate templates for systems with larger chirp masses [4]. Different waveform approximates should produce comparable results, so templates do not need to be generated using the

1Geometric placement algorithms determine which points in the signal parameter space should be included in the template bank to achieve the desired minimum match via analytical methods, thus geometric placement algorithms are only possible for analytical approximates such as TaylorF2. 2Stochastic placement algorithms determine which points in the signal parameter space should be included in the template bank to achieve the desired minimum match by randomly sampling the parameter space. Stochastic placement algorithms are generally slower and more computationally expensive than geometric placement algorithms.

53 same approximate that was used to place them.

Component Mass m1,2 [1, 99] Total Mass M = m1 + m2 [2, 100] Mass Ratio q = m1/m2 [1, ∼98] ˆ NS Dimensionless Component Spin χ1,2 [-0.05, 0.05] L ˆ BH Dimensionless Component Spin χ1,2 [-0.9895, 0.9895] L BNS Region Mass Boundary m1,2 < 2 NSBH Region Mass Boundary m1 ≥ 2, m2 < 2 BBH Region Mass Boundary m1,2 ≥ 2 Lower Frequency Cutoﬀ 30 Hz

Table 2.1. The parameters of the template bank used in the analysis that estimated the signiﬁcance of GW150914.

2.2.2 Likelihood-ratio

The likelihood-ratio used during O1 contained four parameters: the SNR of each trigger involved in the coincidence, the autocorrelation-based test statistic ξ2 of each trigger, the set of detectors involved in the coincidence, and the sensitivities of the detectors. The advanced LIGO detectors were the only detectors operating in O1; it was assumed they were equally sensitive throughout O1, which was a good approximation. Only triggers that were coincident across detectors in template and time were considered. The terms in the factored likelihood-ratio numerator are those in Ref. [46] and shown in Eqs. 1.141 and 1.142. The observatories terms, ~ ~ ~ P O DH, S and P O N , were both unity, as only data from two detectors were available and only candidates that were coincident between the two were considered.

2.3 Initial CBC Results

The initial GW150914 results found after analyzing 507286 seconds, 5.87 days, of coincident data, taken September 12, 2015, 0:00 UTC to September 26, 2015, 3:00 UTC, are shown in table 2.2. The IFAR (inverse False-alarm rate) plots in Fig. 2.1 show the expected and observed inverse False-alarm rates. The closed box results are generated by shifting the timestamps on L1 forward by 8π seconds. We know coincidences formed between a time-shifted detector and the non-time-shifted

54 detector are not gravitational waves, thus the number of observed candidates below a given IFAR threshold should match expectation from noise. The closed box results conﬁrm that the noise model is assigning accurate False-alarm rates. The observed candidates fall below the expected line in the tail of the distribution, but √ still mostly within ± N (1 standard deviation in a poisson process) and generally in the small-number region of the cumulative distribution. The vertical bar in the observed distributions in the closed and open box results at low IFAR is a result of the extinction model described in Sec. 1.6.5. The extinction model used at the time only mapped logL to false-alarm rate above a given logL, anything below that was given the maximum FAR, or minimum IFAR.

2 2 Time FAR (Hz) p-value logL H1 ρ L1 ρ H1 ξ L1 ξ Mass 1 (M ) Mass 2 (M ) Spin-z 1 Spin-z 2 1126259462.4264 2.754e-36 1.671e-30 77.53 20.06 13.46 1.012 0.7314 47.93 36.60 0.9617 -0.8997 Table 2.2. Information about GW150914 from the initial oﬄine GstLAL-based inspiral pipeline analysis, which analyzed 5 days of coincident data for compact binaries in the range listed in Table 2.1.

2.4 Published CBC Results

The final published results came from an analysis of data taken from September 12, 2015, 0:00 UTC to Oct 20, 2015, 13:30:00 UTC. 1507409 seconds (17.45 days) worth of coincident data was analyzed from this time. The analysis was completing by first running the GstLAL-based inspiral pipeline over three seperate chunks of time, 509117 seconds (5.89 days) of data taken from September 12, 2015, 0:00 UTC to September 26, 2015, 3:00 UTC, 533090 seconds (6.17 days) of data taken from September 26, 2015, 3:00 UTC to October 9, 2015, 4:07:11 UTC, and 465202 seconds (5.38 days) taken from October 9, 2015, 4:07:11 UTC to Oct 20, 2015, 13:30:00 UTC. The amount of time analyzed in the first chunk increased from the initial analysis because a new calibration3 was available. The distributions of log L from each chunk are then marginalized over to form one universal mapping from log L to FAR and p-value. The interval of time chosen for this analysis was a collaboration decision. The updated IFAR plots can be seen in Fig. 2.2. Again,

3The detector outputs voltages at the anti-symmetric port, calibration is necessary to convert from units of volts to units of strain. The data available immediately has a uncertainty in the function used to convert from voltage to strain than later calibration versions that take longer to compute.

55 2 2 Time FAR (Hz) p-value logL H1 ρ L1 ρ H1 ξ L1 ξ Mass 1 (M ) Mass 2 (M ) Spin-z 1 Spin-z 2 1126259462.4264 3.270e-36 5.610e-30 77.57 20.08 13.46 1.022 0.7315 47.93 36.60 0.9617 -0.8997 1128678900.4418 1.798e-8 3.039e-2 16.81 7.08 6.80 0.6152 1.022 50.09 8.69 0.2699 -0.1422 Table 2.3. Results from the analysis that the published significance estimate of GW150914 came from [6]. The analysis considered roughly 16 days of coincident data, and looked for signals from compact binary’s whose properties lay in the template bank described in Table 2.1. The first candidate is Gw150914, the second is what is now known as GW151012. At the time, it was not significant enough to warrant a detection claim. The values listed in this table differ from what was published because the GstLAL-based inspiral pipeline’s default noise model is used here, which estimates the distribution of log L in noise using non-coincident triggers. Due no other data producing triggers similar to GW150914 in either detector, the pipeline assigned it an extremely low p-value. The collaboration requested that the noise model also include observed candidates, and published those numbers. GW150914 is so different from the noise of the detectors however that that significance was still greater than 5σ even when it was in its own background. This dissertation will only contain significance estimates from the default noise model, as it minimizes signal contamination. the closed box result comes from shifting L1’s data forward in time by 8π seconds, resulting in a set of coincidences that is not expected to contain gravitational waves. The fact that the observed line follows the expected line in the closed box means that the noise model is assigning accurate FARs. GW150914 was identified with a FAR of 3.270 × 10−36 Hz (1 per ∼ 9.70 octillion, 1027, years) and a p-value of 5.610 × 10−30. The other event that stands out in the open box plot in Fig. 2.2 is GW151012, which was recovered with a FAR of 1.798 × 10−8 Hz (1 per ∼ 1.8 years), with a p-value of 3.039 × 10−2 (1.88σ). Although GW151012 is believed to be a geniune gravitational wave signal now that we have seen several similar events and because the significance has only increased as analysis pipelines have become more sensitive, its significance was not high enough for most to be comfortable confidently calling it a gravitational wave at the time of the first detection, so it was named LVT151012 instead, where LVT stands for “LIGO Virgo Trigger”. GW150914 lives in a region of the likelihood-ratio parameter space where there is no noise; this can be seen in Fig. 2.3, which shows the pipeline’s estimation of P(ρ, ξ2/ρ2| N ) for both the Hanford and Livingston detectors in the warm colormap. The cool colormap is the same distribution, but the candidates found by the pipeline have been added to the histogram before it is converted into a PDF. The cool colormap PDF has been made transparent in areas where it matches the original PDF. The only significant deviations come from the GW150914 trig-

56 gers; the location of GW150914 in this parameter space is shown by a red X, the blue + is the location of GW151012. The PDFs in Fig. 2.3 are the result of marginalizing over all of the background bins; the only triggers that show up in the region near GW150914 are from GW150914. The published signiﬁcance from the GstLAL-based inspiral pipeline was estimated by including zerolag4 candidates in the distribution sampled to produce P (log L| N ). The numbers printed in this thesis do not include zerolag candidates in the estimation of P (log L| N ), resulting in a noise model that has a lower rate of signal contanimation.

2.4.1 p-value sanity check

For Gaussian noise, the distribution of network SNR is a four degree-of-freedom χ2-distribution. Its cumulative distribution, renormalized to only be nonzero for √ √ ρ ∈ [4 2, ∞), where 4 2 is the minimum network SNR that will result in a trigger in the GstLAL-based inspiral pipeline, is

16 e 2 √ P(ρ | N ) dρ = ρ3e−ρn/2dρ , ρ ∈ [4 2, ∞). (2.1) n n 34 n n n

Using Eq. 1.127, the sample rate of 2048Hz, the duration of the analysis, 1507409 seconds, and assuming that every sample drawn is completely independent, the p- value of measuring a candidate with a network SNR of at least 24.17 from Gaussian noise is 6.61×10−110. Assuming that every sample is completely independent when they are not5 provides a lower limit of the p-value. This is a very rough estimate of the p-value; two of its many weaknesses are that we know that the data is not Gaussian, and we know the noise does actually produce candidates with network SNRs as higher and higher than GW150914 from Fig. 2.3. However, if one thinks of the noise as having a Gaussian component and a non-Gaussian glitch component, we can assume the region that GW150914 resides in only consists of Gaussian noise because we can see in Fig. 2.3 that triggers from GW150914 are the sole occupant of that region of parameter space. This calculation of a minimum bound on the p-value assuming Gaussian noise is provided simply as a sanity check of GW150914’s p-value of 5.6 × 10−30 (Table 2.3). The lower-bound is 80 orders of

4Zerolag candidates are coincidences found in non-time-shifted data. 5The whitened data are uncorrelated, however the SNR time series has non-zero correlations between nearby points equal to the autocorrelation of the template.

57 magnitude smaller than GW150914’s estimated p-value.

2.5 Comparison to Parameter Estimation Results

To estimate the parameters of GW150914, the LIGO Scientific Collaboration (LSC) uses two different classes of waveform models: EOBNR and IMRPhenom, both described in Ref. [53]. The EOBNR waveform models assume the component spins are either aligned or anti-aligned with the binary’s angular momentum, which reduces the dimensionality of the parameter space from 15 dimensions to 11 [53]. The IMRPhenom waveform approximate does not make this assumption, instead it models the effect that the 4-extra spin components (Six, Siy, i = {1, 2}) have on the waveform through the lower-dimensional effective spin parameter χp, which is a measure of the precession effects of the binary. χp ∈ [0, 1], where a value of 0 corresponds to a system with completely aligned or anti-aligned spins and hence no precession, and a value of 1 corresponds to the maximum level of precession. Using χp, the IMRPhenom waveform models explore a 13-dimensional parameter space [53]. Table 2.4 lists a side-by-side comparison of the parameters of GW150914 found by the parameter estimation runs and by GstLAL. The parameter estimation process calculates the posterior probability of each of the parameters, meaning that it does not compute a point estimate of the parameters, but instead a PDF of parameters believed to be the most likely. The results are reported as 90% credible-intervals.

The posterior probability of χp closely resembled the prior used to obtain it, indicating that the data does not constrain the possible values of χp very well. The point estimates provided by GstLAL do not fall in any of the 90% credible intervals. This is not surprising, as the bank we used for this analysis is a grid on only 4 parameters: m1, m2, s1z, and s2z. The bank assumes the spins are either aligned or anti-aligned with the binary orbital angular momentum, thus it is exploring a coarsely-grained 4-dimensional subset of the full 11-dimensional parameter space (11 because we assume six = siy = 0 for i = {0, 1}). Although GW150914 was not identiﬁed in low-latency by the GstLAL-based inspiral pipeline, a comparison can still be made between the skymap generated by BAYESTAR [7] (a rapid localization program) estimates using the SNR from the pipeline and the skymap estimated from the full parameter estimation, shown in

58 GstLAL Point Estimate Parameter Estimation 90% Credible Interval +5.6±0.6 m1/M 47.93 38.9−4.3±0.4 +4.2±0.1 m2/M 36.60 31.6−4.7±0.9 +2.1±0.2 M/M 36.40 30.4−1.9±0.5 +4.6±0.5 M/M 84.53 70.6−4.5±1.3 +0.49±0.06 |a1| 0.9617 0.32−0.29±0.01 +0.50±0.08 |a2| 0.8997 0.44−0.40±0.02 Table 2.4. The point estimates of parameters from the template GstLAL-based inspiral pipeline chose as the most signiﬁcant, compared to the results of a full Bayesian analysis that explored a much denser and higher-dimension parameter space but only considers data near the time of the candidate. ai is the dimensionless spin magnitude, which can vary from 0 to 1. The parameter estimation results are in the detector frame, which has been redshifted, making the masses appear larger than they are in the source frame by a factor of (1+z). The template parameters do not fall inside the 90% conﬁdence intervals for any of the masses, and only one of the spins. It is unclear why, though there are possible reasons (e.g. the signal does not contain many cycles to help distinguish it from other waveforms, this may conspire with the fact that the search pipelines only consider a sparse, low-dimensional parameter space to distort the parameter estimates, or to go in the other direction, perhaps the assumption of Gaussian noise by the parameter estimation analyses does not accurately describe the noise as well as the GstLAL-based inspiral pipeline’s noise model; there is no evidence for either hypothesis).

Fig. 2.4. The former’s 90% credible region is 400 deg2, the latter’s is 630 deg2. The area of the overlap between the two skymaps’ 90% credible regions is 360 deg2 [54]. The skymap produced by the parameter estimation pipeline is more accurate as it compares the data to million of waveforms, sampling a parameter space of at least 11-dimensios, while BAYESTAR only considers the times, phases, and amplitudes of the signal [55], however BAYESTAR [7] takes far less time to produce a skymap and is used to produce skymaps rapidly for candidates identiﬁed by CBC matched- ﬁltering pipelines in low-latency.

2.6 Estimated Rates

The rate of merging compact binaries in units of Gpc−3 yr−1 can be estimated from an analysis via Λ R = , (2.2) hVTi

59 where Λ is the number of compact binaries observed in the analysis and hVTi is the estimated space-time volume the analysis is sensitive to averaged over time and a given source population. hVTi is estimated by injecting a population of signals into the data and measuring how many are recovered above a given sig- niﬁcance threshold. The O1-era LVC publications used a Poisson-mixture model, developed independently in Refs. [56, 57], to describe the conditional distribution of the number of noise and signal candidates given the set of ranking statistics of the candidates observed by a search,

P (Λ0, Λ1| {log Lj | j = 1,...,M}) ∝ P (Λ0, Λ1) M Y × exp [−(Λ0 + Λ1)] [Λ0P (log Li| N ) + Λ1P (log Li| S)] (2.3) i=1 where Λ0 is the number of candidates from noise, Λ1 is the number of candidates from gravitational waves from compact binary mergers, M is the number of candidates, {log Lj | j = 1,...,M} is the set of log-likelihood-ratios from the search, and P (Λ0, Λ1) is the prior on the counts. Given a set of candidates with log- likelihood-ratios, Eq. 2.3 is estimated using a Markov Chain Monte Carlo sampler with a Jeffrey’s prior [58]. For the GstLAL-based inspiral pipeline, the median and +7.9 symmetric 90% credible interval for Λ1 were found to be 4.8−3.8 [58]. hVTi was estimated for 4 different source population models [58]. The first source population had dimensionless aligned spins sampled from a uniform distribution over [−.99, 0.99], and masses sampled from a uniform in log distribution for m1,2 ≥ 5M and m1 + m2 ≤ 100M ,

1 P(m1, m2) ∝ , m1,2 ≥ 5M , m1 + m2 ≤ 100M . (2.4) m1m2

The second source distribution had the same spin distribution as the ﬁrst, but it drew the heavier mass from a power law with index −2.35 [58] and the same bounds, −2.35 P(m1) ∝ m1 , m1 ∈ [5M , 95M ] (2.5) The lighter mass was then drawn from a uniform distribution with a minimum mass of 5M and under the constraint that m1 + m2 ≤ 100M . The third and fourth distributions were based on the two most signiﬁcant candidates from the analysis

60 described in Sec. 2.4; the results of these populations will not be discussed here. All of the injected source populations assume sources are uniform in comoving volume and time. Since all of the rates estimates published at the time of writing have included estimates with the power-law and uniform-in-log populations described above, focusing on them makes the evolution of the rates estimate with more detections more clear. The distribution of the volume of space-time in which the analysis is sensitive to gravitational waves is assumed to be log-normal, with a mean set by the log of the average volume found via injections and the standard deviation set by an estimate of the uncertainty in calibration and the uncertainty in the measured average from estimating it from a ﬁnite number of injections. The +0.051 3 estimated hVTi was then 0.080−0.031 Gpc yr for the uniform-in-log population and +0.015 3 0.024−0.009 Gpc yr for the power-law population [58]. Combining the estimated counts and estimated hVTi from above, the median and symmetric 90% credible interval of the rate of binary black hole mergers is +122 −3 −1 60−48 Gpc yr if the source population follow a uniform-in-log distribution and +410 −3 −1 200−160 Gpc yr if the source population follows a power-law distribution for an index of −2.35 [58].

61 Figure 2.1. Plots of the number of candidates with inverse false-alarm rates (IFAR) above the threshold on the x-axis. The top plot is known as a closed box plot, and is from an analysis where the L1 data has been shifted in time so that any coincidences that form between the two detectors are most likely noise. The fact that the observed distribution, the solid line, follows the expected distribution tells us that the false alarm rates being assigned are accurate for noise (i.e. this is veriﬁcation that a false alarm rate of e.g. 1 per day actually means one only expects something at least this signiﬁcant to happen once a day). The bottom plot shows the open box analysis, showing the real results. GW150914 falls far outside the expected region for noise with an IFAR that is 26 orders of magnitude greater than the right edge of the plot. The straight line on the left is due to the analysis only considering candidates above some log L threshold, any events with lower ranking-statistic values are simply assigned the FAR associated with the lowest ranking-statistic value considered.

62 Figure 2.2. The IFAR plots for the analysis that the published GW150914 results are from. The top plot is produced by shifting the L1 timestamps and then analyzing the coincidences normally, providing a check that the FAR estimation is accurate. The fact that the time shifted candidates lie within the expected region means the FARs are accurate for noise. The bottom plot shows the open box results, where data has not been time shifted. There are two candidates that stand out, the marginal candidate GW151012 (which was not granted a GW title until 2018 [5]) and GW150914. The IFAR of GW150914 is 26 orders of magnitude greater than the right edge of the plot.

63 Figure 2.3. The (ρ, ξ2/ρ2) PDF under the noise hypothesis, marginalized across the entire template bank. The PDFs shown in the warm color maps use the GstLAL-based inspiral pipeline’s default noise model, which is estimated from non-coincident triggers from each detector. The regions show in the blue color map are from the PDF estimated using a noise model that includes observed candidates; only regions that are signiﬁcantly diﬀerent from the default noise model PDF are shown. The only regions shown in blue are from GW150914. Part of the procedure to estimate the PDF involves smoothing the histograms, hence the counts from GW150914 are spread out into a few bins and produce the large “protrusion” at the bottom. This is a plot I made, originally for Ref. [6].

Figure 2.4. The skymap estimated rapidly using BAYESTAR [7] and the SNRs found by the GstLAL-based inspiral pipeline (left) compared to the ﬁnal skymap estimated from the parameter estimation runs (right). The overlap region is 360 deg2.

64 Chapter 3 | GW151226

3.1 Introduction

After the discovery of GW150914, the collaboration’s compact binary working group began to consider searching for binary black hole mergers in low latency. After months of discussion, we began uploading low-latency binary black hole merger candidates to GraceDB1 on December 23, 2015. GW151226 was identified within 70 seconds of it passing through the detectors the (UTC) morning of December 26. Finding a second 5σ candidate set many minds at ease, and provided evidence that the first detection was not a statistical fluke. Futhermore, GW151226 is lighter than GW150914 and, unlike GW150914, shows evidence of having non-zero spin. It was now clear that binary black hole mergers were going to become a common occurrence, and increased everybody’s confidence in LVT151012 (though not quite to the point of being comfortable relabeling it to a GW).

3.2 Methods

The template bank and likelihood-ratio are the same as described in Secs. 2.2.1 and 2.2.2. 1GraceDB is the Gravitational-wave Candidate Event Database [10].

65 3.3 Low-Latency Detection

GW151226 passed through the detectors at 03:38:53 UTC December 26, 2015, and was identified in low-latency by the GstLAL-based inspiral pipelineand uploaded to GraceDB [10] at 03:40:00 UTC December 26, 2015 [12]. GW151226 was not detected by the low-latency burst pipelines, and was only identified in low-latency because we started uploading candidates from a low-latency search using the template bank described in Sec. 2.2.1 on December 22, 2015. This analysis began, and hence began collecting statistics to inform the noise model, November 15, 2015. The analysis identified GW151226 with a p-value of 1.12 × 10−4, or 3.7σ, and a FAR of 3.33 × 10−11 Hz. The template parameters are shown in Table 3.1, and the skymap produced by bayestar can be seen in the left-hand column of Fig. 3.2.

2 2 Time FAR (Hz) p-value logL H1 ρ L1 ρ H1 ξ L1 ξ Mass 1 (M ) Mass 2 (M ) Spin-z 1 Spin-z 2 1135136350.6478 3.333e-11 1.120e-4 22.60 9.08 7.39 1.007 1.086 19.92 6.43 0.3396 -0.1239 Table 3.1. The significance and template parameters GW151226 was identified with in low-latency. The analysis began analyzing low-latency data from the detectors on November 15, 2015, and used the template bank described in Table 2.1. GW151226 was identified and uploaded to GraceDB [10] within 70 seconds of passing through the detector.

3.3.1 Multi-Messenger Follow-Up

An alert announcing the identification and localization estimate of GW151226 was distributed to astronomers that had an MOU with the LSC on December 27, 2015 [59]. Although no significant counterparts were identified, 38 different instruments from around the world followed the event up and distributed alert circulars detailing their results [60]. Two neutrinos were found in temporal coincidence with GW151226, though they were not directionally coincident and were otherwise consistent with background [61].

3.4 Published Oﬄine Results

The signiﬁcance from the GstLAL-based inspiral pipeline published in the PRL announcing the detection of GW151226 [12] came from an analysis over all of the coincident data in O1 with an updated calibration relative to the calibration used

66 for the results in Sec. 2.4 and with an updated list of data-quality vetoes from the LSC’s detector characterization group (see Sec. 4 of Ref. [62]). O1 was divided into 3 chunks for this analysis, which was partially motivated by the changing sensitivities of the detectors over the observing run. At this time, the likelihood- ratio did not use any knowledge of the sensitivity of the detectors, so instead of using the noise properties averaged across the entire run to compute the likelihood- ratio, it was computed using local properties by splitting O1 into 3 chunks and analyzing each of these chunks independently. The first chunk contained 1,444,650 seconds (16.72 days) of data taken from September 12, 2015, 0:00 UTC to October 20, 2015, 13:30 UTC. The amount of time analyzed is different from Sec. 2.4 because of improved calibration and the updated list of vetoes. The second chunk contained 1,390,374 seconds (16.09 days) of data taken from October 20, 2015, 13:30 UTC to December 3, 2015, 10:30 UTC. The final chunk contained 1,341,148 seconds (15.52 days) of data taken from December 3, 2015, 10:30 UTC to January 19, 2016, 17:08 UTC. The chunks were then combined using the same combining methodology described in Sec. 2.4. Altogether, 4,176,172 seconds (48.34 days) of data were used to estimate the significance of GW151226. The IFAR plots from this analysis are shown in Fig. 3.1. The closed box results are again generated by adding 8π seconds to the L1 data before analyzing, leading to a collection of candidates that are unlikely to contain a true gravitational wave signal. The loudest event in the right plot in Fig. 3.1 is GW150914, the second loudest is GW151226. The parameters that GW151226, including the FAR and p-value, are shown in Table 3.2. As described in Sec. 2.4, the published results include zerolag candidates when computing P (log L| N ) (excluding the previously identified GW150914), while the results shown here minimize signal contamination by excluding zerolag candidates in the calculation. Using this noise model, a log L of 33.7 corresponds to a p-value of 1.77 × 10−9, or 5.90σ, and a FAR of 3.63 × 10−16 Hz, or 1 per 87.3 million years.

2 2 Time FAR (Hz) p-value logL H1 ρ L1 ρ H1 ξ L1 ξ Mass 1 (M ) Mass 2 (M ) Spin-z 1 Spin-z 2 1135136350.6493 3.63e-16 1.77e-9 33.7 10.51 7.59 0.8588 1.106 19.75 6.62 0.4742 -0.3591 Table 3.2. The significance estimate and parameters of the best-fit template for GW151226, taken from the analysis that produced the published significance estimate [12]. As described below Table 2.3, the noise model used in this dissertation is the GstLAL-based inspiral pipeline’s default noise model that minimizes signal contamination, which is different than the noise model used in the paper.

67 The significance estimate varied between the final published value and the initial low-latency value because they are fundamentally different analyses. The calibration of the data is different, the template binning (see Sec. 1.6.1) is different, and the livetime analyzed are different. The analyzed times change for a few reasons. First of all, the low-latency analysis analyzes all data that is denoted as “science quality," although sometimes the detector only has a few seconds of science quality data at a time. These times do not get analyzed offline because we require a minimum of 512 seconds of continuous science quality data for each interval analyzed. Additionally, the offline analysis only considers times where both detectors have science quality data. At the time of writing, only non-coincident triggers that occur during times when coincidences are possible (i.e. more than one detector has science quality data) are used to inform the noise model, however during O1 all non-coincident triggers were used to inform the noise model, so the online analysis analyzed a lot of data that were not analyzed offline. These times were included in later analyses, see Ref. [63]. The offline analysis also vetoed times said to have poor data quality by the LSC’s detector characterization group. Moreover, the offline analysis should be more sensitive to gravitational waves, as it is able to analyze all of the specified data to gather noise statistics before ranking candidates and assigning significances while the online analysis only has the data available before the time of a candidate to inform its knowledge of the noise.

3.5 Comparison to Parameter Estimation Results

The source properties of GW151226 were estimated in the same way as GW150914 and using the same waveform families, EOBNR and IMRPhenom (see Sec. 2.5). The results for mass and spin, published in Ref. [8], are listed in Table 3.3 alongside the point estimated from the GstLAL-based inspiral pipeline. The skymap estimated by LALInference is shown in the right column of Fig. 3.2, alongside the original skymap generated by bayestar using the details from the low-latency detection. The point estimate are all within 90% of the conﬁdence intervals of the parameter estimation results. Measuring closer parameters than in the case of GW150914 is not unexpected, as GW151226 spent about one second in the detector’s sensitive frequency band, compared to GW150914 which only spent about 200 ms. Thus there are around a factor of 5 times as many sample points to

68 compare between the data and the templates.

GstLAL Point Estimate Parameter Estimation Median and 90% Credible Interval +9.0±2.6 m1/M 19.75 15.6−5.0±0.2 +2.6±0.2 m2/M 6.62 8.2−2.5±0.5 +0.07±0.01 M/M 9.68 9.72−0.06±0.01 +6.5±2.2 M/M 26.37 23.7−1.4±0.1 +0.37±0.11 |a1| 0.4742 0.49−0.42±0.07 0.43±0.01 |a2| 0.3591 0.52−0.47±0.00 Table 3.3. The parameters of the template the GstLAL-based inspiral pipeline used to identify GW151226 compared to the detector-frame median and 90% conﬁdence intervals estimated by parameter estimation pipelines. All of the point estimates fall within the 90% conﬁdence intervals. GW151226 was in the detector’s sensitive frequency band for longer than GW150914, which may explain the improvement in parameter estimation.

3.6 Estimated Rates

The rates estimated in Sec. 2.6 were updated using all of the O1 analysis described in Sec. 3.4, using the same methodology. For the uniform-in-log-mass distribution, the new median and symmetric 90% credible interval estimated from the output +43 −3 −1 of the GstLAL-based inspiral pipeline was 29−21 Gpc yr [8], while the previous +122 −3 −1 estimate was 60−48 Gpc yr . For the power-law distribution with an index of - +137 −3 −1 2.35, the new values were found to be 94−66 Mpc yr [8], while it was previously +410 −3 −1 estimated to be 200−160 Gpc yr .

69 Figure 3.1. The expected counts above a given inverse FAR (IFAR) threshold for the analysis the published GW151226 significance estimate was taken from. This analysis was over all of the coincident data taken during O1, using the template bank described in Table 2.1. The closed box plot (top) is generated the same way as the previous closed box plots (see e.g. Fig. 2.2) and shows that the FAR calculation produces accurate FARs for noise. The open box plot (bottom) cuts off at too low of an IFAR to see GW151226 or GW150914. The third loudest candidate in O1 was GW151012, then named LVT151012 to reflect its marginal significance.

70 Figure 3.2. The original skymap estimated by BAYESTAR [7] using the low-latency detection of GW151226 (left), and the ﬁnal skymap from a full Bayesian analysis (right).

71 Chapter 4 | GW170104

4.1 Introduction

After O1, the two LIGO detectors decided to try implementing separate planned upgrades. L1’s horizon distance increased by a factor of 4/3 at the beginning of O2, however there were diﬃculties in the H1 upgrades and its sensitivity stayed approximately the same as at the end of O1 [5,13]. O2 began November 30, 2016 and took data until December 23, when it shut down for a short period of time so that researchers would not need to work through the holidays like they did when GW151226 was detected. The observing run resumed January 4, 2017, and shortly thereafter GW170104 passed through the detectors. It was not detected by the low-latency pipeline due to metadata associated with low-latency data from H1 being set incorrectly. At time of its discovery, GW170104 was believed to be the most distant gravitational wave source seen.

4.2 Methods

4.2.1 Template Bank

As was done for the O1 banks (Sec. 2.2.1), the template bank used in the first published analysis of GW170104 was created using two different waveform families and two different placement algorithms. This bank had a minimum match of 97%. Binary systems with a total mass between 2 M and 4 M were generated with the TaylorF2 approximate again [38], and initial placement was done using a geometric method [3]. Further placement was done stochastically using

72 the geometrically placed bank as a starting point. Templates with larger total masses, up to M = 400 M , were generated using an updated version of the effective one body formalism approximate used before (Sec. 2.2.1), referred to as SEOBNRv4_ROM [41]. The suﬃx, ROM, stands for reduced order model; ROM waveforms are computationally cheaper and faster to generate, and result in errors smaller than those introduced by data calibration [41]. The SEOBNRv4_ROM waveforms were placed used a stochastic method [3]. The parameters of the bank are shown in Table 4.1, and a plot of the bank in (m1, m2) coordinates can be seen in Fig. 4.1.

Component Mass m1,2 [1, 399] Total Mass M = m1 + m2 [2, 400] Mass Ratio q = m1/m2 [1, ∼98] ˆ NS Dimensionless Component Spin χ1,2 [-0.05, 0.05] L ˆ BH Dimensionless Component Spin χ1,2 [-0.999, 0.999] L BNS Region Mass Boundary m1,2 < 2 NSBH Region Mass Boundary m1 ≥ 2, m2 < 2 BBH Region Mass Boundary m1,2 ≥ 2 Lower Frequency Cutoﬀ 15 Hz Minimum Duration 200 ms

Table 4.1. The parameters of the template bank used by the analysis that produced the published GW170104 signiﬁcance estimate.

Figure 4.1. A plot of the component masses of the template bank used by the analysis that produced the published GW170104 signiﬁcance estimate, originally published in Ref. [3].

73 4.2.2 Likelihood-ratio

The likelihood-ratio used in this analysis includes two new parameters relative to the likelihood-ratio used in the initial O1 analyses (Sec. 2.2.2), the diﬀerence in end times measured at each detector and the diﬀerence in coalescence phases measured at each detector, ∆t and ∆φ. One of my contributions to the GstLAL-based inspiral pipeline was devising the mathematics of adding two new parameters to the likelihood-ratio, developing a method to compute one of the two new terms, and implementing the new terms in the analysis. The work is summarized in Ref. [9]. We focused on creating a likelihood-ratio for the H1L1 network for simplicity, though work was later done to generalize the likelihood-ratio to N detectors. The numerator and denominator of the likelihood-ratio are factored identically to Eqs. 1.141 and 1.142, with the only dependence on ∆t and ∆φ appearing in the term shown below,

~2 ~ ~ L ~ρ, ξ , DH, O, ∆tH1L1, ∆φH1L1 ~ ~ P(~ρ, ∆tH1L1, ∆φH1L1| DH, O, S = · · · × , (4.1) P (∆tH1L1, ∆φH1L1| N ) where it is assumed that ∆t and ∆φ are not correlated with any of the other variables under the noise hypothesis, and only correlated with SNR and the horizon distances under the signal hypothesis. The instrument term, O~ is unnecessary since we are only considering the H1L1 network, so it will be dropped for the rest of the chapter. Under the noise hypothesis, we assume that ∆t and ∆φ are independent,

P (∆tH1L1, ∆φH1L1| N ) = P (∆tH1L1| N ) P (∆φH1L1| N ) . (4.2)

In the absence of signal there should be no preferred ∆t or ∆φ, thus we model both of these distributions as uniform. Under the signal hypothesis, the term in the numerator shown above can be factored as

~ ~ P(~ρ, ∆tH1L1, ∆φH1L1| O, S = P (~ρ| DH, S ~ × P (∆tH1L1| ~ρ, O, S P (∆φH1L1| ~ρ, ∆tH1L1, S) , (4.3)

74 where we have assumed that ∆φ is only weakly correlated with the detector sensitivities and thus ignore the correlation. The ﬁrst term in Eq. 4.3 is unchanged from before adding ∆t and ∆φ. For the second term, we assumed that the majority of the correlation between ∆t and the SNRs and sensitivitives can be captured by a new variable,  ρ /D  H1 HH1  ρ /D ρH1/DHH1 ≤ ρL1/DHL1 , ρ ≡ L1 HL1 (4.4) ratio ρ /D  L1 HL1  ρ /D ρH1/DHH1 > ρL1/DHL1 , H1 HH1 which is deﬁned to always be in the interval (0, 1]. The ratio of SNR to horizon distance can be thought of as a normalized SNR, if both horizon distances are the same you expect the SNRs to be similar (though it depends on where the source is with respect to each detector’s antenna pattern). Additionally, the higher the SNR in a single detector, the better the measurement resolution of the end time of the signal is, i.e. we are able to more accurately measure the end time of louder signals.

Combining these two ideas, ∆t when ρratio is near unity should have a diﬀerent structure than when ρratio is small. For the third term in Eq. 4.3, we assume that the majority of the correlation between SNRs and ∆φ can be described using the network SNR, ρnw, in place of the two individual SNRs from H1 and L1. Thus,

~ ~ P(~ρ, ∆tH1L1, ∆φH1L1| DH, S ≈ P(~ρ| O, S

× P (∆tH1L1| ρratio, S) P (∆φH1L1| ∆tH1L1, ρnw, S) . (4.5)

Note the approximate relationship in Eq. 4.5 was not rigorously proven, though the estimated Volume-Time (VT) did increase as a result of this work.

We estimated P (∆tH1L1| ρratio, S) and P (∆φH1L1| ∆tH1L1, ρnw, S) using distributions measured from the injection campaign to estimate the binary black hole rates in O1 (Sec. 3.6). My work was focused on rapidly estimating the ∆t signal distribution, P (∆tH1L1| ρratio, S), while a post-doctoral researcher at University of

Wisconsin Milwaukee focused on rapidly estimating P (∆φH1L1| ∆tH1L1, ρnw, S).

1The light-travel-time from H1 to L1 is a little less than 10 ms, the coincidence window is 5 ms larger than the light travel time to account for time-measurement errors.

75 a Gaussian kernel and normalized the histograms so that the resulting PDFs smoothly tapered to 0 after |∆t| = 10 ms. The results for each ρratio bin were then fit to 12-degree Chebyshev polnomials with even parity (all of the odd coefficients are zero), which were experimentally found to sufficiently approximate the distributions. Examples for three separate ρratio bins are shown in Fig. 4.2. Finally, the coefficients of the Chebyshev polynomials were fit to polynomials that were a function of ρratio, such that an approximation of the ∆t distribution could be rapidly constructed for arbitrary ρratio. The predicted normalized distributions for the ρratio bins shown in Fig. 4.2 are shown in Fig. 4.3. This fit was found to approximate the distributions well for ρratio ≥ 0.33; there were not enough injections for smaller values of ρratio to accurately measure the distribution. For a

2-detector network, the majority of detections should be above ρratio = 0.33, thus any candidate that had ρratio < 0.33 used the ∆t distribution at ρratio = 0.33. The ∆φ signal distributions were first binned by H1 and L1 SNRs, then the sum of a Von Mises distribution and a uniform distribution were fit to the normalized histograms. I attempted to apply a similar methodology to this, using Chebyshev polynomials and then a polynomial fit to predict the Chebyshev coeff- cients, but results were not as accurate at the Von Mises distribution. The ∆t and ∆φ distributions derived from this work are shown in Fig. 4.4.

4.3 Published Oﬄine Results

The signiﬁcance estimate of GW170104 published in Ref. [13] is from an analysis of 478,575 seconds (5.54 days) of data taken from January 4, 2017, 00:00 UTC to January 22, 2017, 08:00 UTC. This is shorter than previous analyses due to a perceived decrease in pressure by the collaboration to reach 5σ certainty; previously ∼ 15 days had been used because another compact binary analysis pipeline used by the LSC determined it needed at least 15 days of coincident data to collect enough background statistics to reach a 5σ certainty, though the GstLAL-based inspiral pipeline does not share this constraint. The IFAR plots are shown in Fig. 4.5. As before, the noise model is estimated from non-coincident triggers, though starting in O2 the noise model was estimated exclusively using non-coincident triggers which could have formed a coincidence, i.e. triggers from times when only one detector was operating were not used to

76 Figure 4.2. ∆t distributions under the signal hypothesis, estimated from the analysis of injected binary black hole mergers used to estimate merger rates at the end of O1 in Ref. [8]. The unsmoothed data is a histogram, the smoothed data is then the histogram smoothed with a Gaussian kernel, and the ﬁt is computed using Chebyshev polynomials. inform the noise model. The closed box result is once again from shifting the data from L1 in time and analyzing the result, which minimizes signal contamination in the closed box result. GW170104 was identiﬁed with a p-value of 1.295 × 10−7 (5.15σ), and a FAR of 2.136 × 10−13 Hz (1 per ∼ 150,000 years). The second loudest candidate in the open box has a p-value of 0.082 (1.39σ) and a FAR of 1.415 × 10−7 Hz (1 per ∼ 82 days), and thus is not inconsistent with noise. The parameters of the two loudest events are shown in Table 4.2, though it should be noted that subsequent analysis over all of O1 and O2 did not identify the second loudest event as being inconsistent with noise.

77 Figure 4.3. A sanity check of the numerical method used to compute the ∆t signal- model term in the likelihood-ratio. The line labeled “original fit” is a Chebyshev polynomial computed using coefficients that were fit to the observed data. The predicted fit line is the result of using Chebyshev polynomials to estimate the Chebyshev coefficients used in the original fit as a function of ρ.

2 2 Time FAR (Hz) p-value logL H1 ρ L1 ρ H1 ξ L1 ξ Mass 1 (M ) Mass 2 (M ) Spin-z 1 Spin-z 2 1167559936.6014 2.136e-13 2.136e-13 27.98 8.31 9.87 0.992 0.844 28.06 24.10 0.0761 -0.9013 1168667947.8025 1.415e-7 8.224e-2 15.38 5.59 7.63 0.480 1.087 47.34 1.06 -0.7099 0.0268 Table 4.2. The significance estimates and parameters of the two most significant candidates found by the analysis that produced the published GW170104 significance estimate [13]. The loudest candidate is GW170104, the second loudest occurred at 05:58:49 UTC on January 17, 2017, but subsequent analyses substantially decreased its significance and thus it is believed to be noise. As explained in Table 2.3, the noise model used to estimate the significance estimates listed here is the default used by the GstLAL-based inspiral pipeline which minimizes signal contamination, while the noise model used to estimate the published results used a noise model that added observed candidates to the noise.

4.4 Comparison to Parameter Estimation Results

The parameter estimation runs once again used two waveform families: IMRPhe- nomPv2, which incorporates precession effects via the effective precession param- 78 Figure 4.4. The ∆t and ∆φ signal models used in the likelihood-ratio numerator, computed using models whose coefficients have been fit to the distributions observed in the population of injections used to estimate binary black hole merger rates in O1. Figure from Ref. [9].

eter, χp, and SEOBNRv3, which attempts to model two-spin effects more generally [13]. The final results were averaged across the results from runs using these two waveform families, and are shown alongside point estimates from the GstLAL- based inspiral pipeline in Table 4.3. The point estimates of the chirp mass and component spins are within the 90% confidence intervals; however, the point estimates of the component masses and the total mass are not. The chirp mass is much easier to measure than individual component masses as it drives the behavior of the inspiral portion of the waveform at the leading order, thus it is not surprising that the component masses and total mass are not measured as well. As was the case with GW151226, the signal spent more time in the detector’s sensitive frequency band than GW150914; the best-match template for GW170104 is ∼ 3.7 seconds long2. Even though GW170104 was not found by the low-latency GstLAL-based inspiral pipeline, data from the time of the event were analyzed after the fact using the offline configuration with background statistics collected by the low-latency analysis up to that point. This analysis was performed the day after, so the background is not what it would have been and the statistical statements have no valid meaning, thus they will not be reported here. Furthermore, only a small subset of

2Recall that the template GW170104 was identiﬁed by has a lower starting frequency than the templates GW150914 and GW151226 were identiﬁed by.

79 GstLAL Point Estimate Parameter Estimation Median and 90% Credible Interval +9.0±1.2 m1/M 28.06 36.7−6.8±0.4 +6.2±0.2 m2/M 24.10 22.9−7.5±0.1 +2.5±0.2 M/M 22.63 25.1−3.9±0.4 +5.7±0.1 M/M 52.16 59.9−6.5±1.0 +0.46±0.02 |a1| 0.0761 0.45−0.40±0.01 +0.46±0.01 |a2| 0.9013 0.47−0.43±0.01 Table 4.3. A comparison of the parameters from the template the GstLAL-based inspiral pipeline associated with GW170104 and the median and 90% credible region detector-frame parameter estimates from the parameter estimation pipelines, which implement a full Bayesian analysis. The chirp mass and the spin magnitude estimates are within the 90% confidence regions, though the component masses and total mass are not. To zeroth order though, the behavior of the waveform is proportional to the chirp mass, thus if only one parameter is estimated correctly it is expected to be the chirp mass. the template bank was used. The purpose of this work was to ensure both compact binary matched-filtering pipelines identified GW170104. The skymaps are shown in Fig. 4.6.

4.5 Estimated Rates

The rates estimates were updated with the analysis described here, as well as an analysis of data taken earlier in O2 which did not identify any high conﬁdence events. This analysis considered 501,107 seconds (5.80 days) of data taken from November 30, 2016, 16:00 UTC (the beginning of O2) to December 23, 2016, 00:00 UTC, the start of the holiday break. The same populations were used as in Secs. 2.6 and 3.6. The updated median and 90% conﬁdence interval for the uniform-in-log- +33 −3 −1 mass population are 32−20 Gpc yr ; for the power-law population with an index +110 −3 −1 of −2.35 the updated numbers are 103−63 Gpc yr [13]. This is consistent +122 −3 −1 with the estimates using all of O1, 60−48 Gpc yr for the uniform-in-log-mass +410 −3 −1 population and 200−160 Gpc yr for the power-law population [8].

80 Figure 4.5. The counts above a given IFAR threshold vs the IFAR threshold. The closed box results (top) were generated using the method described in Fig. 2.2 and should be interpreted in the same way, i.e. we know the analysis is assigning accurate FARs to noise candidates because the observed closed box results lie within the expected region. The open box (bottom) show two candidates that appear to stick out above noise, GW170104, which has an IFAR 3 orders of magnitude higher than the right edge of the plot , and a 1.4σ candidate that disappeared in subsequent analyses.

81 Figure 4.6. The skymap estimated by BAYESTAR [7] using SNRs from the GstLAL- based inspiral pipeline is on the left, and the skymap estimated from the ﬁnal parameter estimation run which performs a full Bayesian analysis is on the right. The data used to generate the BAYESTAR skymap has never been released publicly, and in fact did not come from an analysis over the full template bank; the data are the result of a sanity check that was done where templates near the expected GW170104 parameters were matched-ﬁltered and the output was ranked using the state of the low-latency background at the time of the analysis (roughly a day after GW170104).

82 Chapter 5 | GW170814

5.1 Introduction

The advanced Virgo detector joined O2 on August 1, 2017. Two weeks later, gravitational waves from a binary black hole merger were observed by three ground- based detectors for the first time. GW170814 passed through the detectors at 10:30:43 UTC and was identified in ∼ 30 seconds. GW170814 is particularly important in the history of gravitational wave astronomy for 3 distinct reasons: first, it was the first time an instrument not built or maintained by the LIGO Lab participated in a gravitational wave detection, adding further credibility to the existence and successful detection of gravitational waves; second, the triple-coincidence detection provided evidence that the Virgo detector was functioning as intended, evidence that would be crucial only three days later when the lack of a loud signal in Virgo would drastically improve the localization of a gravitational wave [11] (Ch. 6); third, the two LIGO detectors are approximately co-aligned and thus are sensitive to the same polarization and insensitive to the orthogonal polarization, but the Virgo detector is not co-aligned with the two LIGO detectors and thus provides additional polarization information not measurable by the LIGO detectors alone [14].

83 5.2 Methods

5.2.1 Template Bank

Two separate template banks were used in the GW170814 analyses: one for the low-latency analysis, and a different one for the offline analysis. Both template banks were modified from the template bank described in Sec. 4.2.1.

5.2.1.1 Low-latency Bank

The bank used in the low-latency analysis is identical to the bank used in the GW170104 analysis (Sec. 4.2.1), except that it has a total mass upper cutoﬀ of

150 M [3]. This is because the binning procedure used for the low-latency analysis, based on chirp mass and the effective spin parameter, χ, binned waveforms together from systems with very different total masses. This was a problem because systems with large total masses are more susceptible to glitches due to their small number of cycles and short duration. This mass-based susceptibility broke a key assumption in the GstLAL-based inspiral pipeline, namely that templates in a given background bin have similar responses to noise. This caused a net drop in sensitivity in high mass regions, and may have caused the assigned FARs to be inaccurate [3]; this may be the reason why the marginal candidate discussed in Sec. 4.3 disappeared in subsequent analyses, though this is only speculation. This bank is shown in Fig. 5.1. For a summary of its properties, see Table. 4.1, though note that this bank has an upper-total-mass cutoff of 150 M .

5.2.1.2 Oﬄine Bank

In order to fix the problems described in Sec. 5.2.1.1, more templates were added to the bank described in Sec. 4.2.1 and the binning algorithm for determining which templates are grouped together for background bins was modified. Templates with total masses larger than 80 M from the bank described in Sec. 4.2.1 was first used as a seed for stochastically adding templates to raise the minimum match to 0.98 from 0.97. The duration cut of 200 ms was also dropped during this stage. After the additional templates were added to increase the minimum match, 1000 templates with total masses between 100 M and 400 M were added by hand in

84 Figure 5.1. The component masses of the template bank used in the low-latency analysis that detected GW170814 and GW170817 (Ch. 6). The template bank is identical to the template bank described in Table 4.1 except that it has a total mass upper cutoff of 150 M . This figure originally appeared in Ref. [3]. a uniform grid in component mass. These were added to make it easier to group templates with other templates that have similar responses to noise in the binning process. The final result is plotted in terms of component mass in Fig. 5.2. Grouping the templates in this bank using the chirp mass and χ binning still resulted in templates in the high mass region with different noise properties being grouped together. We found that binning templates with M < 80 M using the original method and binning higher total-mass templates by template duration solved this problem. More information about this problem and the solution can be found in Ref. [9].

5.2.2 Likelihood-ratio

The low-latency analysis that detected GW170814 analyzed times when only one detector had science quality data. Single detector candidates were ranked using the same likelihood-ratio, though in the single detector case there are no ∆t or ∆φ terms. Additionally, the O~ term in the likelihood-ratio (Eqs. 1.141 and 1.142) is now relevant, and can be {H1}, {L1}, or {H1, L1}, representing single-detector

85 Figure 5.2. The template bank used by the analysis that computed the published GW170814 significance. It is a modified version of the bank described in Table 4.1. The high mass region now has a minimum match of 0.98, up from 0.97. 1000 additional templates were then added to the high mass region in a grid that is uniformly spaced in component mass. This figure originally appeared in Ref. [3].

H1 candidates, single-detector L1 candidates, or coincident H1L1 candidates. The analysis also matched-ﬁltered data from Virgo, however Virgo was not incorporated into the likelihood-ratio. This had the net eﬀect that Virgo did not affect the analysis, but trigger information from Virgo was uploaded to GraceDB [10] when available and coincident with a candidate from H1 and/or L1; the information from Virgo was used by BAYESTAR [7] to compute skymaps.

In the oﬄine analysis, the calculation of the horizon distances, DHH1 and DHL1 , changed from a point estimate of the horizon distance from the time of the event to a volume-weighted estimate over the previous 5 minutes. More details are discussed in Sec. 6.2.1.

5.3 Low-Latency Detection

The analysis that detected GW170814 began analyzing data on June 15, 2017. It analyzed data from both LIGO detectors and the Virgo detector, which joined O2

86 on August 1, 2017. As stated above, Virgo data was not incorporated into the analysis’s signiﬁcance estimates, though if Virgo SNR passed the trigger threshold a trigger was generated and used to inform BAYESTAR [7] if uploaded to GraceDB [10]. The most signiﬁcant low-latency candidate associated with GW170814 was uploaded to GraceDB by the GstLAL-based inspiral pipeline ∼ 32.5 s after the gravitational waves passed through the detectors with a p-value of 5.501 × 10−6 (4.40σ) and a FAR of 3.830 × 10−13 Hz (1 per ∼ 83,000 years). The parameters of this candidate are shown in Table 5.1. The skymap from this upload is shown in Fig. 5.4.

2 2 2 Time FAR (Hz) p-value logL H1 ρ L1 ρ V1 ρ H1 ξ L1 ξ V1 ξ Mass 1 (M ) Mass 2 (M ) Spin-z 1 Spin-z 2 1186741861.5268 3.830e-13 5.501e-6 34.37 7.25 13.74 4.39 1.1769 1.0761 0.9543 29.48 24.90 -0.5688 0.1308 Table 5.1. The significance and template parameters of the most significant low-latency candidate associated with GW170814. At the time of detection, Virgo triggers were generated to check for coincidence but were not incorporated into the likelihood-ratio, i.e. the Virgo trigger did not affect the significance estimate. It did, however, improve the localization [14]. The noise model used to estimate the significance estimated here is again the default noise model that minimizes signal contamination as opposed to the noise model used to estimate the published significance estimates which includes observed candidates, as described in Table 2.3.

At this time, we had a parallel analysis running that did not consider data from Virgo; significance estimates from that analysis will not be discussed here. The first candidate associated with GW170814 uploaded to GraceDB was uploaded by this low-latency analysis ∼ 16.5 s after the gravitational waves passed through the detectors. The difference in latencies comes from the Virgo calibration having a higher latency than the LIGO calibration.

5.3.1 Multi-Messenger Follow-Up

An alert for GW170814 was distributed to astronomers that had signed an MOU approximately 2 hours after the initial identification of GW170814 [64]. Altogether, 14 different instruments participated in follow-up of GW170814 and distributed circulars [65], though no significant counterparts were identified.

87 5.4 Published Oﬄine Results

The significance estimate published in Ref. [14] came from an analysis of 508,387 seconds (5.88 days) of H1, L1 data, taken from August 13, 2017, 02:00 UTC to August 21, 2017, 05:00 UTC. V1 data was not considered offline because it does not affect the significance estimate and the offline search pipeline results are not used to generate new skymaps; the final skymaps come from the parameter estimation pipelines, which did incorporate V1 data. As with previous detections, the published numbers added event candidates into the noise model, while the numbers this thesis presents use a noise model that does not include event candidates, minimizing signal contamination. Furthermore, the numbers published in Ref. [14] use a different livetime to compute the FAR than the analysis uses by default. By default, the analysis uses timestamps taken from the output stream buffers generated by the GStreamer element that takes SNR time-series in and outputs triggers. Effectively this means that the livetime is the accumulated wall clock time each analysis was run over, in the low-latency analysis this means that the livetime used to normalize FARs is approximately the wall clock time during which the analysis has been running, but offline this means the livetime will differ from the length of time where both detectors have coincident science-quality data. For example, if there is a short period of time in the middle of a segment being analyzed where one detector does not have science quality data while the other does, both detectors will still have non-zero livetime throughout this interval. The values reported in this dissertation use the definition of livetime derived from GStreamer buffers as described above, while Ref. [14] uses the duration of science-quality segments as its livetime. The p-value of GW170814 was 1.431 × 10−17 (8.45σ), and the FAR was 2.452 × 10−23 Hz (1 per 1.292 × 1015 years). The parameters of the best-fit template are shown in Table 5.2. The IFAR plots are shown in Fig 5.3. The closed box analysis is over data where L1 was shifted forward in time relative to H1. GW170814 is the second loudest event in the open box, the loudest is GW170817 (Ch. 6), though both have FARs that are too low to be shown on the default IFAR plots produced by the analysis.

88 2 2 Time FAR (Hz) p-value logL H1 ρ L1 ρ H1 ξ L1 ξ Mass 1 (M ) Mass 2 (M ) Spin-z 1 Spin-z 2 1186741861.5283 2.452e-23 1.431e-17 50.87 6.91 13.87 1.3244 0.9966 50.04 21.33 0.5229 -0.5053 Table 5.2. The significance and template parameters of the most significant candidate associated with GW170814, as determined by an analysis of roughly 5 days of coincident LIGO data. Virgo data were not used in the significance estimate, though they were used for rapid localization [7] and parameter estimation. As Virgo data are not used to estimate significance, they were not included in the offline analysis, and hence there is no Virgo SNR or ξ2. The noise model used to estimate the significance estimated here is the same as was used previously, which differs from the noise model used to estimate the published significance, see Table 2.3.

5.5 Comparison to Parameter Estimation Results

The same waveform approximates were used to estimate the source properties of GW170814 as were used for GW170104 (Sec. 4.4), only a newer version of the family which tries to generally account for 2-spin dynamics was used, speciﬁcally SEOBNRv4_ROM [41]. Results from the two families were averaged to produce the numbers published in Ref. [14]. The detector frame numbers were not published, but they can be computed from the posterior samples available to collaboration members. These numbers are shown next to the point estimates from GstLAL-based inspiral pipeline in Table 5.3. Starting with this detection, individual spin values were not published, instead the eﬀective spin parameter χ was published to convey the information inferred about the source system’s spins. The

GstLAL Point Estimate Parameter Estimation Median and 90% Credible Interval +6.3 m1/M 50.04 33.8−3.2 +3.0 m2/M 21.33 27.0−4.6 +1.3 M/M 27.94 26.7−1.2 +3.4 M/M 71.36 62.0−2.8 +0.12 χ 0.2156 0.06−0.12 Table 5.3. The parameters of the template associated with the most signiﬁcant GW170814 candidate as reported by the GstLAL-based inspiral pipeline compared to the detector-frame median and 90% credible intervals found by the parameter estimation results. The chirp mass is the only template parameter that falls within the 90% credible region found by the parameter estimation analysis, though the leading order behavior of the inspiral waveform scales with chirp mass so that is not surprising. point estimate of the chirp mass is the only point estimate that falls in the ﬁnal

89 90% confidence interval. As in the case of GW170104 (Sec. 4.4), this is not surprising as the chirp mass dictates the leading order behavior of the waveform. The duration of the most significant template associated with GW170814 is ∼ 2.7 s. Additionally, the data used in the parameter estimation analyses were different than the data used in significance estimate. The data used to estimate the properties of the source had known noise sources subtracted [14], while the data used to estimate the significance did not. The low-latency skymap and the final skymap from the parameter estimation runs are shown side by side in Fig. 5.4. The low-latency map was found to have a systematic shift relative to the final skymap. This was later tracked to the zero- latency whitener used in the pipeline at this time [66]. When using that whitener, differences in the PSD used to whiten the templates and the measured PSD at the time of the candidate can shift the timestamps of the output from one detector. At the time of writing, the zero-latency whitener is no longer being used, though future work will adjust the output timestamps to account for this shift.

5.6 Estimated Rates

The collaboration’s rate estimates were not updated using GW170814, primarily because it was found close to the end of the second observing run and a paper recomputing the rates based on all of O1 and O2 was already planned. These updated rate estimates are discussed in Ch. 7

90 Figure 5.3. The number of candidates above an IFAR threshold plotted against the IFAR threshold. The closed box results (top) are generated and should be interpreted the same way as the closed box results described in Fig. 2.2. Once again, the fact that the observed closed box counts lie within the expected region is evidence the analysis is assigning accurate FARs. There are two signals in the open box plot (bottom), both to the right of the visible region: GW170814, which has an IFAR 13 orders of magnitude larger than the right edge of the plot, and GW170817, which has an IFAR 21 orders of magnitude larger than the right edge of the plot.

91 Figure 5.4. The initial skymap is on the left, which was estimated rapidly [7] using the low-latency detection. A bug was later identified which resulted in a systematic shift of the skymap by a small amount, though it has since been fixed. The final skymap is on the right, estimated by a full Bayesian analysis.

92 Chapter 6 | GW170817

6.1 Introduction

It was widely believed that binary neutron star (BNS) mergers would be the most plentiful source of gravitational waves observable with ground-based detectors. They were also a more exciting detection prospect for non-gravitational-wave astronomers, because it has long been believed that BNS mergers were short Gamma- Ray Burst progenitors. Although it hadn’t been long enough to confidently say the BNS merger rate was lower than the expected rate, many astronomers were beginning to ask if they should be worried that no BNS mergers had been detected yet. The first detection of gravitational waves from a binary neutron star merger initially went unnoticed by most. Although low-latency candidates uploaded to GraceDB [10] below a set FAR threshold normally started a mass internal alert that notified collaboration members, the LIGO Scientific and Virgo collaborations had a policy during O2 that they would only distribute alerts for “extraordinary” single- detector gravitational wave candidates. Members of the GstLAL development team, such as myself and Chad Hanna, set up our own internal alerts for single- detector candidates, and as a result were notified when GW170817 was uploaded GraceDB as a single-detector H1 candidate. At the time of the upload, a sGRB observed by Fermi-LAT, now named GRB 170817A, had already been uploaded to GraceDB as an external event and was coincident in time with GW170817. With the help of Reed Essick, we were able to trigger the internal alert system to notify the collaborations that we had just made the first joint multi-messenger detection

93 in 30 years and had long sought-after evidence that binary neutron star mergers were short GRB progenitors.

6.2 Methods

GW170817 was identified as a single-detector candidate instead of a coincident H1 L1 candidate1 because there was a loud glitch, a common term for transient noise, in L1 shortly before merger which caused the LSC’s automatic data-quality algorithms to veto the L1 trigger. For reasons explained in the likelihood-section below, this resulted in the analysis believing L1 was off at the time of GW170817. A patch to the likelihood-ratio code from the development version the GstLAL library was backported to solve this problem in the offline analysis of GW170817. The same template bank and likelihood-ratio were used in the low-latency and offline analyses of GW170817 as were used for GW170814 (Sec. 5.2).

6.2.1 Likelihood-ratio

Up to this point in O2, the horizon distance estimation used in the likelihood-ratio as a measure of the sensitivity of the detectors was a point estimate from the time of the candidate. This resulted in the analysis registering L1 as off at the time of GW170817, as the glitch just before merger caused a data-quality veto to flag the data as bad which set DHL1 to zero. To fix this, code was backported from the GstLAL development branch to estimate the average volume-weighted horizon distance over the five minutes prior to an event, but only in the offline analysis code. The low-latency analysis did not incorporate this change. In the offline analysis, the horizon distance for each detector is now

Z t 1/3 1 0 0 3 DH(t) = dt (DHinst (t )) , (6.1) 300 t−300

where DHinst (t) is the instantaneous horizon at time t, which is computed using the PSD at time t when not at a vetoed time and zero during vetoed times. Cubing the horizon distance returns a term proportional to the volume of space

1GW170817 was near one of Virgo’s blind spots, and thus was not loud enough to produce a trigger in Virgo [11]

94 the detector is sensitive to, and integrating that value over time gives a term proportional to the VT observed by the detector over the previous ﬁve minutes. Dividing by the total time of the integral returns a term proportional to the time- averaged volume the detector was sensitive to, and taking the cube root of that returns the volume-weighted average distance to which the detector is expected to observe an optimally oriented system with an SNR of 8. The integral in Eq. 6.1 is implemented numerically as a sum. The diﬀerence between the instantaneous horizon distance and the volume-weighted average can be seen in Fig. 6.1, where the time was chosen such that H1 data did not have many vetoes while L1 data did. The volume-weighted average is insensitive to the occasional veto, while times that where many of the data are vetoed are correctly measured to be less sensitive. The new procedure results in a decreased horizon distance at the beginning of a segment of science-quality data, but gravitational waves and glitches can occur at any time in a segment while the beginning only happens once for each segment, thus fewer signals are likely to be missed using this procedure than the original. Work done after this analysis changed the average so that it is computed over the length of the template instead of the arbitrarily-chosen 5 minutes.

6.3 Low-Latency Detection

GW170817 was detected by the low-latency GstLAL-based inspiral pipeline and uploaded to GraceDB [10] at 12:47:18 UTC on August 17, 2017. It was ﬁrst uploaded as a single-detector candidate with a p-value of 5.089 × 10−5 (3.886σ) and a FAR of 3.478×10−12 Hz (1 per ∼ 9,000 years). The template parameters are shown in Table 6.1. This template had a duration of ∼ 360 s. The collaboration

2 Time FAR (Hz) p-value logL H1 ρ H1 ξ Mass 1 (M ) Mass 2 (M ) Spin-z 1 Spin-z 2 1187008882.4457 3.478e-12 5.089e-5 14.45 14.45 1.865 1.53 1.24 -0.0159 -0.0357 Table 6.1. The signiﬁcance estimate and template parameters associated with the most signiﬁcant low-latency GW170817 candidate, which was found as a single-detector H1 candidate. had a policy at the time that single detector candidates would not be considered for generating private alerts sent to astronomers that signed agreements with the LIGO Lab and the LSC, but I had alerts on GraceDB set up to text me if we uploaded any candidates with a FAR below in 1 in 30 days. We quickly saw that there was a

95 track visible in the time-frequency plot of data from H1 surrounding the time of the candidate (Fig. 6.2(a)), consistent with what we expected from a binary neutron star merger. Additionally, a short GRB observed by Fermi roughly 2 seconds after the merger had been uploaded to GraceDB at 12:41:45 UTC. After alerting the collaboration, we found out that the Virgo data transfer from the detector in Italy to the cluster at the California Institute of Technology had crashed at 10:44:50 UTC. It was also discovered that a track was also visible in the time-frequency plot of data from L1, behind a large noise transient that was obviously a glitch (Fig. 6.2(b)); the glitch triggered a detchar veto flag, thus our point estimate of the horizon distance was set to zero for L1 and the analysis then believed L1 was offline. The first skymap generated, shown in Fig. 6.3(a), was a single detector skymap and thus only shows the antenna pattern at the time of the merger. Another skymap was generated hours later using SNR from all three detectors after the glitch was excised [11], shown in Fig. 6.3(b).

6.3.1 Multi-Messenger Follow-Up

The first alert about GW170817 was sent to astronomers that had signed an MOU about 40 minutes after it passed through the detectors [67]. 59 different groups, not counting the LIGO Scientific and Virgo collaborations, participated in the follow-up of GW170817 [68]. A bright optical transient believed to be a kilonova was identified in a known galaxy, NGC 4993, less than 9 hours later, and was subsequently observed by multiple teams and instruments. Over the next several weeks, electromagnetic radiation was observed from that region in the ultraviolet, optical, infrared, X-ray, and radio bands; the ultraviolet, optical, and infrared are believed to be from the kilonova, while the X-ray and radio emissions are afterglows from the GRB. The discovery of the kilonova confirmed BNS mergers as kilonovae progenitors in addition to short GRB progenitors, and confirmed that short GRB can be seen by off-axis observers and in those cases kilonovae can also be observed. This may drastically increase the rate at which coincident gravitational-wave and short GRB detections may be made. Further observations of the kilonova found evidence of nuclear decay of r-process elements, elements heavier than and including iron that cannot be produced in the

96 same manner as as elements lighter than iron. This supports the model that BNS mergers lead to r-process nucleosynthesis, the production of r-process elements, and that these mergers may be a significant source of heavy elements in the universe [68, 69]. The identification of NGC 4933 as the host galaxy provided the redshift of a gravitational wave source independent of gravitational wave data for the first time [68]. It also proved that GRB 170817A was both the dimmest and closest short GRB observed with a known redshift to date. The implication of this discovery is two-fold: first it proves that short GRBs can be observed off-axis, second it suggests that a subset of the short GRBs that have been observed with unknown redshifts may be closer than previously thought. Separate from the GRB, combining the redshift with the luminosity distance estimate from GW170817 provides a new measurement of the Hubble constant, a measurement of the rate of expansion +12.0 −1 −1 in the local universe, of 70.0−8.0 km s Mpc [70]. There are other measurements of the Hubble constant, but currently there is significant tension between two which disagree with each other at the 3σ level [70]. The first of these two is 67.8±0.9 km s−1 Mpc−1 from measurements of the Cosmic Microwave Background (CMB) [71]; the second is 73.24 ± 1.74 km s−1 Mpc−1 and is estimated using su- pernovae [72]. The new measurement of the Hubble constant using GW170817 and the redshift of NGC 4993 agrees with both of these measurements, but the uncertainty will decrease as more BNS mergers with known redshifts are observed. The beginning of GRB 170817A was found to occur 1.74 ± 0.05 s after the peak strain of GW170817 [73]. This was used along with the luminosity distance estimates from GW170817 to set upper and lower bounds on the difference between the speeds of light and gravity. The upper bound comes from assuming the peak gravitational-wave strain was emitted at the same time as the start of the GRB, and the lower bound comes from assuming the GRB was emitted 10 s after the peak strain [73]. The difference between the speeds of light and gravity was measured

−15 vGW−c −16 to be −3 × 10 ≤ c ≤ 7 × 10 [73]. These bounds were then used to test gravitational Lorentz Invariance, a fundamental principle that essentially states the same laws of physics hold in every locally ﬂat region of space-time2. The bounds on the diﬀerence in the speeds of light and gravity were also used

2It should be noted that locally flat in general relativity means flat at an infinitesimal point, not necessarily in the neighborhood of the point.

97 to test whether gravitational and electromagnetic waves are equally affected by gravitational potentials as predicted by general relativity. This is checked using the Shapiro effect [74], which states that massless particles (e.g. photons and gravitons) should take slightly longer to propagate through curved space-times compared to flat space-times. This delay is parameterized by the dimensionless γ, and according −5 to general relativity γGW = γEM = 1. Using the bound γEM −1 = (2.1±2.3)×10 from Ref. [75] and the bounds on the difference in the speeds of light and gravity −7 −6 from above, −2.6 × 10 ≤ γGW − γEM ≤ 1.2 × 10 [73]. The results discussed here are a summary of the work in Refs. [68,73], though new papers studying this multi-messenger event are still being published today, and new observations of the radio afterglow are still being made nearly 2 years after the discovery.

6.4 Published Oﬄine Results

The published estimated significance of GW170817 came from the same analysis as the published GW170814 numbers, discussed in Sec. 5.4 and Ref. [14]. Using the noise model that minimizes signal contamination by not including candidates and using our default definition of livetime (Sec. 5.4), GW170817 was found with a p-value of 1.645 × 10−25 (10.37σ) and a FAR of 2.819 × 10−31 Hz (1 per 1.124 × 1023 years). The template parameters of the most significant candidate associated GW170817 are shown in Table 6.2. The bad ξ2 value for L1 can be explained by

2 2 Time FAR (Hz) p-value logL H1 ρ L1 ρ H1 ξ L1 ξ Mass 1 (M ) Mass 2 (M ) Spin-z 1 Spin-z 2 1187008882.443 2.819e-31 1.645e-25 68.54 16.22 24.74 0.8396 2.086 1.78 1.08 0.0415 0.0415 Table 6.2. The significance and template parameters of the most significant candidate associated with GW170817. As described in Table 2.3, the noise model used to estimate the significance here is different than the noise model used to produce the published significance, but this noise model is the default for the GstLAL-based inspiral pipeline and minimizes signal contamination. the glitch and a bug that was identified in the application of vetoes that resulted in more time being vetoed. Before GW170817, the detchar group only ever flagged an integer number of seconds as vetoed. They decided to start flagging data that should be gated using their approved methodology on sub-second intervals. During this analysis, we discovered that there was a bug where data would only be gated

98 in integer second intervals. As a result, all of the data at 12:41:03 UTC were gated, when only the data within 0.03125 s of 12:41:03.40625 UTC, plus some padding, should have been (Fig. 6.4). The vetoed times include the region where GW170817 passes through 1024 Hz, thus even after ﬁxing the bug in gated times, the measured ξ2 is expected to be large because the data containing the end of the waveform are gated, thus the SNR samples immediately surrounding that end of the template will not match the autocorrelation very well.

6.5 Comparison to Parameter Estimation Results

The parameter estimation results published in the GW170817 detection paper [11] were obtained using the TaylorF2 waveforms based on the post-Newtonian approximation. The TaylorF2 waveform used describes the point-particle interactions of the binary up to post-Newtonian order 3.5, and the phase up to post-Newtonian order 5, where tidal effects enter via the Love number, Λ [11,76,77], and assumes the spin angular momentum is aligned or anti-aligned with the orbital angular momentum. Due to a degeneracy between the ratio of component masses and the component spins, parameters were estimated under two different spin populations. The first population, motivated by the observed Galactic population of binary neutron stars expected to merge within a Hubble time [11, 34], consists of individual spin magnitudes not larger than 0.05. The second population, motivated by the contemporary understanding of the neutron star equation of state, consists of individual spin magnitudes not larger than 0.89 [11]. This bound was set by the technical limits of rapidly and efficiently generating waveforms, though most equations of state do not support component spins above 0.7. The parameter estimation runs began at 30 Hz, above which GW170817 had ∼ 3000 cycles [11], making the parameter estimation runs much more computationally expensive and time-consuming than the previous analyses, which considered binary black hole merger candidates. The large number of cycles in the inspiral stage of the waveform allows us to constrain the chirp mass for GW170817 much more precisely than for all previous events, and the point estimation from the GstLAL-based inspiral pipeline agrees with the parameter estimation results. The component masses, total mass, and effective spin parameter are within the 90% confidence intervals estimated when spins were allowed to be as large as 0.89, though they are

99 GstLAL Point Estimate Parameter Estimation Results (Low Spin) Parameter Estimation Results (High Spin)

m1/M 1.78 1.38 − 1.62 1.37 − 2.28

m2/M 1.08 1.18 − 1.38 0.87 − 1.37 +0.0002 +0.0008 M/M 1.1977 1.1976−0.0002 1.1977−0.0003 +0.04 +0.48 M/M 2.85 2.76−0.01 2.84−0.09 χ 0.0334 −0.01 − 0.02 0.00 − 0.17 Table 6.3. Comparison of the template parameters associated with the most significant GW170817 candidate identified by the GstLAL-based inspiral pipeline and the detector- frame parameters determined by a full Bayesian analysis [11]. The chirp mass is listed to more significant digits than in previous detections because GW170817 has far more cycles than any of the binary black hole merger signals, allowing us to constrain the chirp mass much more than in previous detections. The component mass ranges are the 90% confidence region that includes the equal mass case (see Fig. 6.5), due to the posterior distributions showing substantial support for an equal mass system. There is a large degeneracy between the mass ratio, q = m1/m2, and the effective spin parameter, χ, so two different spin priors were used: a low-spin prior motivated by spins of observed binary neutron stars, and a high-spin prior loosely motivated by physical models and set by technical limitations. The point estimates from the GstLAL-based inspiral pipeline fall within the 90% credible region of the high-spin priors, though the chirp mass estimate agrees with the range estimated using both spin priors. not within the 90% confidence intervals estimated when spins were restricted to be less than 0.05. The 90% credible intervals listed for m1 and m2 begin/end end with the equal mass case. The results are given in this format because there is nonzero support for a mass ratio of one, which can be inferred from the component mass posteriors from the detection paper, Ref. [11], as shown in Fig. 6.5. The dashed lines in Fig. 6.5 show the lower (higher) bound of the 90th percentile for m1 (m2), starting at equal masses.

6.6 Estimated Rates

The rate of binary neutron star mergers published in the GW170817 detection paper [11] was estimated using a method diﬀerent than the method used to estimate the binary black hole merger rates prior to this event (Secs. 2.6, 3.6, and 4.5). The primary motivation to use a diﬀerent method to estimate the rate was that the collaboration did not have time to run the full analyses required for the prior method, the joint-detection of electromagnetic counterparts meant that the collaboration had to work with other collaborations when making decisions on when to

100 announce, and the external astronomy community generally has a lower cadence between discovery and publication than the LSC. The method used in Ref. [11] is ﬁrst outlined in Ref. [78], though instead of observing zero candidates it was assumed we saw one. Following the derivation in Ref. [78] but using the likelihood of observing one signal instead of observing zero signals, the rate marginalized over hVTi becomes

Z P(R| d) ∝ R dhVTi P(R| hVTi)P(hVTi) hVTie−RhVTi, (6.2) where in this context, d is the observed data. An exponential prior was used for the rate, P(R| hVTi), with a scale parameter set by the 90% upper-limit of the rate estimate from O1 [11, 78]. The prior on hVTi, P(hVTi), was assumed to be log-normal with shape parameters set by the estimation of hVTi and its uncertainty from both measurement and calibration uncertainties. As was done in the previous method, hVTi was measured by injecting simulated signals in the data and recording how many are recovered. Instead of waiting for an improved data calibration3 to use for published results, the original calibration, with an uncertainty of 30%, was used for the published significance estimates and the standard injection sets used during O2 were used to estimate hVTi. During O2, simulated signals were injected to estimate the sensitivity of the GstLAL-based inspiral pipeline in every analysis performed. Several injection sets were used for each analysis chunk, covering the four most likely sources: BNS mergers, NSBH mergers, binary black hole (BBH) mergers, and intermediate mass binary black hole mergers. Each category also had at least two injection populations, from different waveform approximation families and/or different source parameter distributions. Altogether, there were 11 different injection populations used in the initial O2 analyses. The traditional way to use these populations in an analysis is to inject each population into the data and then analyze the data using the entire template bank, as one would in an analysis without injections4. The template bank used in O2 had 677,000 templates [3], so analyzing the data using the entire template bank

3Improved in this context means to have lower uncertainty. 4There are two key diﬀerences between an “injection" analysis and a standard analysis. First, the background estimate is taken from the standard analysis so that injections don’t contaminate the background. Second, only data surrounding the known injection times are analyzed.

101 11 times is computationally expensive and time-consuming. Fortunately, we can predict to which templates may ring up based on the chirp mass of the injection. I wrote a program named gstlal_inspiral_split_injections that will take any number of injection populations, combine them into one large population, sort the injections by chirp mass, and then write new injection files to disk with an associated chirp mass interval for each file in which to search for file’s injections. For O2, this reduced the number of input injection files from 11 to 2, where one of the files contained all of the low-mass injections and was only analyzed using the low-mass portion of the template bank, while the other file contained all of the high-mass injections and was only analyzed using the high-mass portion of the template bank. The only difference between the files gstlal_inspiral_split_injections takes as input and the original population files used for injections previously is that gstlal_inspiral_split_injections requires that the combined population of injections have staggered end-times (i.e. the same amount of average time must pass between each injection in the combined population). Staggering the end times reduces the number of injections, but also reduces the number of times the data has to be re-analyzed over the entirety of the template bank. The number of files written to disk at the end depends on the rate of injections specified by the user. During O2, we set the target rate to be half that of the input. In other words, if the combined population of injections had a rate of 1 injection every 10 seconds, then the target injection rate gstlal_inspiral_split_injections attempted to reach was 1 injection every 20 seconds. Before writing each new injection file to disk, gstlal_inspiral_split_injections iterates through all of the injections about to be written to disk, keeping a running tally of the predicted largest and smallest recovered chirp masses. Global maximum and minimum chirp masses were kept for each output file, and for each injection’s chirp mass, M, the maximum was taken between the previous maximum and

 1.3M, M < 10   Mupper = 1.5M, 10 ≤ M < 20 , (6.3)   2M, M ≥ 20

102 while the minimum was taken between the previous minimum and

 0.65M, M < 10 Mlower = . (6.4) 0.5M, M ≥ 10

The resulting maximum and minimum chirp masses defined the region of the template bank used to search for injections in the associated file. Eqs. 6.3 and 6.4 were not quantitatively derived, but qualitatively estimated by examining how well the GstLAL-based inspiral pipeline recovered injection parameters. To test that searching for injections using the methods described above were sane, a comparison performed between the two methods was performed using a binary neutron star template bank. The original analysis, which used unal- tered injection files, ran 1,700 different instances of jobs matched-filtering the data around injections, while the analysis which used the injection files produced by gstlal_inspiral_split_injections only ran 300 different instances. The original found 9,830 injections and missed 37,509; the analysis using my program’s injection files found 9,788 injections (< 0.5%) and missed 37,551 injections. The final VT estimates came out within 1% of each other, which can be seen in Fig. 6.6.

There were two binary neutron star injection sets used in O2, both drawn from the same population but with different random seeds and generated using different waveform approximations. The component masses are drawn uniformly between 1 M and 3 M , the chirp-distance is drawn from a uniform distribution from 5 Mpc to 300 Mpc, and the component-spins are isotropic with a magnitude less than 0.4. The only difference between the populations is that the waveforms were generated with different approximations, SpinTaylorT2threePointFivePN and SpinTaylorT4threePointFivePN [38]. The hVTi used in this rates analysis was estimated by only considering injections with component masses between 1 M and 2 M , motivated by the observed distribution of binary neutron stars in the Galaxy. With the estimated hVT i from measuring how many of these injections were recovered in the GstLAL-based inspiral pipeline, and folding in both the measurement and the calibration uncertainties, and using the 90% upper-limit of 12600 Gpc−3 yr−1 found in O1 [78] as a prior, the median and 90% confidence inter-

103 +2920 −3 −1 vals of the distribution deﬁned in Eq. 6.2 were found to be 1420−1150 Gpc yr . The published result comes from combining this posterior with another rate posterior estimated using hVTi from a diﬀerent compact binary pipeline [11].

104 Figure 6.1. A comparison of the original (black line) and updated (dashed red line) horizon distance calculations. The original method computed the horizon distance using the PSD at the time of the candidate, which meant that when a detector characterization data-quality ﬂag vetoed the glitch in L1 just before merger of GW170817, the horizon distance for L1 was set to zero and the low-latency software pipeline treated L1 as oﬀ. The updated algorithm computes a volume-weighted average over the previous 5 minutes, so that short vetoes will not prevent detections. This new algorithm is robust to occasional bursts of noise, and instead returns a lower sensitivity during periods when the glitch rate is high, which is a more desirable behavior.

105 (a) (b)

Figure 6.2. The time-frequency plots automatically generated when GW170817 was uploaded to GraceDB [10]. A faint track is visible in H1, and a much louder track is visible in L1 behind a “glitch,” a common name for transient noise. The glitch is the reason GW170817 was identiﬁed as a single-detector candidate.

(a) (b)

Figure 6.3. The original skymap estimated by BAYESTAR [7] (left), and the 3 detector skymap estimated by BAYESTAR (right). The original skymap simply shows the antenna pattern of H1 at the time of the detection, while the 3-detector skymap had a 90% conﬁdence area of 31 deg2.

106 Figure 6.4. Whitened strain data from L1 at the time of the glitch that occurred before the merger of GW170817. The “Veto Gate” refers to the GStreamer element removes vetoed times, the “Autogate” refers to the algorithm that self-vetoes times where whitened strain data passes a set threshold [4]. The full second of data at 12:41:03 UTC was removed instead of only removing data with 0.03125 s (plus some padding) of 12:41:03.40625 UTC. This was ﬁxed in subsequent analyses, for now the only result is a decreased L1 SNR and a worse ξ2.

107 1.4

1.3 χz < 0.05 | | χz < 0.89 1.2 | |

] 1.1

2 1.0 m 0.9

0.8

0.7

0.6 1.25 1.50 1.75 2.00 2.25 2.50 2.75 m1 [M ]

Figure 6.5. The posterior component mass distributions for GW170817, for both the low-spin prior (motivated by the observed binary neutron star population) and the high- spin prior. Both priors show good support for equal component masses. This ﬁgure originally appeared in Ref. [11].

(a) (b)

Figure 6.6. The average space-time, hVTi, volume the detectors are sensitive to is estimated by injecting simulated signals and recording how the time-averaged fraction that are recovered as a function of distance. The plot on the left shows the estimates using the original injection analysis algorithm. The plot on the right shows the estimate obtained when only looking at speciﬁc regions of the template bank for injections. These regions need to be large enough to include the majority of templates expected to ring up from an injection; I developed an algorithm that chose what region of the template bank to use when searching for analyses based on the parameters of the injections that estimates an average space-time volume accurate to within 1% of the calculation that uses the entire template bank while running in less time and using less computational resources.

108 Chapter 7 | GWTC-1

7.1 Introduction

After two observing runs and multiple detections, the collaboration decided to compile a catalog of known gravitational wave detections and marginal candidates that could be elevated to a detection by multi-messenger follow-up. This catalog, it was decided, would estimate the rate of binary black hole mergers, binary neutron star mergers, and neutron star black hole binary mergers using all of the coincident data collected over O1 and O2; provide updated estimates of the signiﬁcance of all of the detections so far using the best available calibration; and provide updates of the estimated parameters for all of the candidates observed so far using the latest parameter estimation analysis methods. The data from O2 was scheduled to go public in 2019, and this was the collaboration’s last chance to publish results that could be used by other researchers without competing with other gravitational wave astronomers. In addition to providing updates on existing detections, 4 new binary black hole mergers were announced in the catalog paper: GW170729, GW170809, GW170818, and GW170823. These four detections bring the total binary black hole merger detections at the end of O2 to 10 (including GW170608, discussed below).

7.2 Methods

The catalog paper analyses used the same template bank as was used in oﬄine GW170814 and GW170817 analysis, described in Sec. 5.2.1. The GstLAL-based

109 inspiral pipeline replaced the ∆t and ∆φ distributions estimated from O1 injections (Sec. 4.2.2) with a new framework that is generalized to N detectors, enabling us to use Virgo in signiﬁcance estimates.

7.2.1 Likelihood-ratio

The likelihood-ratio used for the catalog paper analyses used a new method of evaluating the probability of the observed extrinsic parameters under the signal hypothesis, specifically the set of observed SNRs, the differences in end times, and the differences in coalescence phases. The new implementation allowed the inclusion of Virgo data in significance estimates during times when it was operating. The end time of the candidate itself was also included as a parameter, where the end time of a coincidence is defined as the end time of the trigger from the first instrument sorted alphabetically. For example, a coincident H1L1 candidate would use the H1 trigger’s end time. The numerator of the likelihood-ratio is now factored as

~2 ~ ~ ~ ~ ~ ~ ~ ~ ~ P ~ρ, ξ , DH, t, φ, O S = P DH P O DH, S P(t0| O, DH, S Y 2 ~ ~ ~ ~ ~ −4 × P ξIFO ρIFO, S P ∆Deﬀ , ∆t, ∆φ O, DH, S |~ρ| , (7.1) IFO∈O~ where t0 is the time of the ﬁrst trigger sorted by observatory alphabetically, and

~ Deff,1 Deff,2 ∆Deff ≡ { , , ···} (7.2) Deff,0 Deff,0 ~ ∆t ≡ {t1 − t0, t2 − t0, ···} (7.3) ~ ∆φ ≡ {φ1 − φ0, φ2 − φ0, ···} (7.4) where the indices label the order of the observatory in the alphabetically sorted list of observatories involved in the detection. The SNR signal distribution is known to be proportional to |~ρ|−4 [79, 80], so the network SNR is factored out and we drop our explicit dependence on the horizon distances in the extrinsic parameters term by switching from considering SNR consistency to considering effective distance consistency across detectors. The extrinsic parameter joint PDF is then approximated by a non-physical numerical method described in Ref. [79],

110 which requires computing many terms in advance, though the calculation only needs to be done once for each set of detectors (for example, the distribution will need to be recomputed before data from Kagra [81] can be analyzed). The other new term in Eq. 7.1 is deﬁned as

!3 D P(t | O,~ D~ , S = Hnet N (7.5) 0 H 150 Mpc tbin where Ntbin is the number of templates in the bin the best-fit template is found in and DHnet is the horizon distance of the least sensitive detector in a network of the minimum number of detectors required to produce a candidate (e.g. the second shortest horizon distance when only considering candidates where at least two detectors are coincident). Note that this term is not normalized, as the normalization is a function of time and thus would constantly change in the low-latency analysis. There are two consequences of including this non-normalized term: first, a likelihood-ratio of 1 no longer means that the observed parameters are equally likely under the noise and signal hypotheses, second, likelihood-ratios can now be compared across analysis boundaries. Before adding this term, candidates could only be compared across analysis boundaries via the p-value or FAR. Not normalizing the term also allows us to not exactly solve for it, instead we can use a term proportional to the probability density that we want. The network horizon distance is divided by 150 Mpc to set the scale of this term, and the cubed result is proportional to the volume of space in which the network of detectors is sensitive to signals. This term is multiplied by the number of templates in the best-fit template’s bin, a quantity that is proportional to the number of trials in that bin. The denominator of the likelihood ratio is now factored as

~2 ~ ~ ~ ~ ~ ~ ~ ~ Y 2 P ~ρ, ξ , DH, t, φ, O N = P DH P O N P t O, N P ρIFO, ξIFO N . IFO∈O~ (7.6) The uniform ∆t and ∆φ terms from before have been dropped, thus the denominator is not normalized, though we no longer care about normalization for the reasons listed above when discussing the numerator. The new term in the denominator is estimated by tracking the trigger rates in each detector and using those rates to compute the coincidence rate. The trigger rates are tracked on a 10 s cadence and are averaged over the hour before and after the candidate to account for sudden

111 changes in trigger rates caused by vetoed data.

7.3 Published Results

The catalog paper, Ref [5], re-analyzed all of the coincident data taken during O1 and O2 using the best calibration1, covering September 12, 2015, 00:00:00 UTC to January 19, 2016, 17:07:59 UTC for O1 and November 30, 2016, 16:00:00 UTC to August 26, 2017, 00:00:00 UTC. The O2 data used also had several independent noise contributions removed, resulting in an 18% gain in the BNS merger range at H1, and a negligible difference at L1 [5]. These data were divided into chunks consisting of roughly 5 days of coincident science-quality H1L1 data, and analyzed in those chunks. Updates to the extinction model since the O1 results (Sec. 3.4) meant that the noise statistics, specifically P (log L| N ), from these runs could not be combined to re-assign FARs and p-values in the same way. The extinction model used during O2 depended on modes in P (log L| N ), which varied from analysis chunk to analysis chunk, thus the extinction model could not be applied to the combined set. Instead, a log L threshold was chosen that was sufficiently high to be above the log L mode in all analysis chunks while still lower than thousands of noise events; every candidate below this log L threshold of 14 was dropped, and the FARs and p-values were then reassigned using an estimate of P (log L| N ) from all of O1 and O2 [5]. The total livetime of this analysis was 16663279.813329 seconds, or about 193 days. As was the case for GW170104, GW170814, and GW170817, the numbers published in the collaboration paper, Ref. [5], use a livetime defined to be the total coincident livetime after subtracting vetoed times, while the numbers that will be provided here use the livetime internally computed by the GstLAL-based inspiral pipeline, which is essentially the wall-clock time of each analysis. To compare between the results, one only needs the two different livetimes: 16663279.813329 seconds and 14662116.992500 seconds. Four new binary black hole detections were announced in Ref. [5], GW170729, GW170809, GW170818, and GW170823. In that order, the estimated FARs and p-values were 5.070 × 10−9 Hz (1 per ∼ 6.25 years) and 8.101 × 10−2 (1.40σ)

1The best calibration is deﬁned to be the calibration with the smallest calibration error. For O1, this is the same calibration used in Refs. [8, 12, 78].

112 for GW170729, 1.278 × 10−21 Hz (1 per ∼ 25 trillion years) and 2.130 × 10−14 (7.55σ) for GW170809, 1.162 × 10−12 Hz (1 per ∼ 27,000 years) and 1.953 × 10−5 (4.11σ) for GW170818, and 1.283 × 10−19 Hz (1 per ∼ 250 billion years) and 2.138 × 10−12 (6.93σ) for GW170823. The template parameters associated with the most significant candidate for each detection can be seen in Table 7.1. The updated FARs and p-values for the previously announced detections are (in order of discovery date): GW150914, 3.069 × 10−47 Hz (1 per ∼ 1.03 duodecillion (1039) years) and 5.114 × 10−40 (13.19σ); GW151012, 2.209 × 10−10 Hz (1 per ∼ 140 years) and 3.674 × 10−3 (2.68σ); GW151226, 5.448 × 10−23 Hz (1 per ∼ 580 trillion years) and 9.078×10−16 (7.95σ); GW170104, 6.960×10−17 Hz (1 per ∼ 450 million years) and 1.160×10−9 (5.97σ); GW170814, 3.473×10−27 Hz (1 per ∼ 9.12 quadrillion years) and 5.787 × 10−20 (9.07σ); GW170817, 2.201 × 10−58 Hz (1 per ∼ 140 quindecillion (1048) years) and 3.668 × 10−51 (15.00σ). The template parameters associated with the most significant candidate for each discovery are listed in Table 7.1. The table is sorted by candidate significance, and includes 14 additional candidates not discussed above. These 14 candidates have FARs below 1 in 30 days, the threshold used in Ref. [5] to determine whether to include marginal candidates. This work has 3 candidates in addition to Ref. [5], due to the different livetimes used to estimate the FARs. Given the analysis time of 16663279.813329 seconds, the Poisson probability of seeing 14 candidates from noise with FARs below 1 per 30 days is 3.814×10−3 (2.67σ). Note that most of the marginal candidates have p-values near unity, thus it is unlikely that a marginal candidate can be elevated to a detection without a multi-messenger counterpart. The IFAR plot from an analysis with more than a one signal event is not very helpful to look at as it becomes difficult to discern if the noise model is working correctly. To get around this problem, a new type of plot was created, shown in Fig. 7.1. Instead of plotting the IFAR of all of the candidates, the ranking statistic (the log-likelihood-ratio for the GstLAL-based inspiral pipeline) is plotted with the noise model, signal model, and combined noise+signal models overlaid. This plot shows that the distribution of ranking statistics follow the distribution expected when sampling from noise populations that dominate in the low-ranking-statistic regime and signal populations that dominate in the high-ranking-statistic regime. The closed box results are also overlaid to show that the noise model correctly predicts the rate of noise events. The figure in the right column is the same

113 Figure 7.1. The number of candidates above a given log-likelihood-ratio threshold vs that threshold. We are now in the regime where the signal population is loud enough that the observed cumulative counts follow a combined noise-and-signal population, thus the signal model and the noise-plus-signal are plotted in addition to the noise model, to check if the observed distribution is as expected. The right plot is zoomed in at the left edge of the left plot, to show that the closed box results (obtained and interpreted the same way as in Fig. 2.2) are in agreement with the noise model. The analysis has become sensitive enough that it can now detect loud signals even in the closed box, pairing the loud signal with a random peak. As a result, the top two loudest closed box candidates were associated with GW170817 and GW150914, respectively, and had to be manually removed to produce a signal-free population to check the noise-model. The shaded regions can be thought of as the “1ish” and “2ish” σ regions. The ﬁgure on the left is a ﬁgure I created for Ref. [5].

figure, zoomed in to show that the time-shifted results fall within the noise error bars. The error bars were computed numerically using the posterior samples from the rates estimate (Sec. 7.5) for the signal model and using the GstLAL-based inspiral pipeline’s estimate of the mean rate of noise events for the noise-model, and approximately represent the ±1σ and ±2σ regions. These regions are not exactly the 1σ or 2σ regions because the signal and noise populations are assumed to be Poisson processes with the Poisson rate at a given ranking statistic given by the y-value in the plot, and for most Poisson rates there is no combination of integer events that will fall exactly on the 1 or 2 σ boundaries. There are two things worth discussing about the results in Table 7.1. Firstly, despite being able to estimate significance using all three detectors, GW170814 was still identified as a double-detector candidate instead of a triple detector. This can happen when a signal is marginal in one of the detectors, noise can push the

114 2 2 2 Label Time FAR (Hz) p-value logL H1 ρ L1 ρ V1 ρ H1 ξ L1 ξ V1 ξ Mass 1 (M ) Mass 2 (M ) Spin-z 1 Spin-z 2 GW170817 1187008882.443 2.201e-58 3.668e-51 134.68 19.11 26.91 0.696 1.626 1.78 1.08 0.0415 0.0200 GW150914 1126259462.430 3.069e-47 5.114e-40 110.05 20.31 13.54 0.906 0.766 46.89 32.33 0.755 -0.875 GW170814 1186741861.526 3.473e-27 5.787e-20 65.24 9.14 13.04 1.230 1.046 59.14 15.23 0.132 0.726 GW151226 1135136350.649 5.448e-23 9.078e-16 55.80 10.51 7.82 0.845 1.110 20.19 6.57 0.232 0.779 GW170809 1186302519.746 1.278e-21 2.130e-14 52.81 6.59 10.53 1.121 0.573 47.12 23.61 0.0329 0.146 GW170823 1187529256.518 1.283e-19 2.138e-12 48.35 6.40 9.52 1.052 0.702 42.44 33.60 -0.697 -0.156 GW170104 1167559936.601 6.960e-17 1.160e-9 42.13 8.94 9.42 0.936 0.969 55.19 15.90 0.268 -0.411 GW170818 1187058327.080 1.172e-12 1.953e-5 32.45 4.13 9.66 4.23 1.073 0.963 0.739 55.38 24.09 0.098 -0.498 GW151012 1128678900.445 2.209e-10 3.674e-3 27.38 7.33 6.76 0.603 0.996 31.73 13.65 0.357 -0.913 GW170729 1185389807.328 5.070e-9 8.101e-2 24.42 7.90 7.34 1.082 0.887 79.20 49.12 0.457 0.344 170405 1175425510.693 1.269e-7 8.793e-1 21.39 4.66 8.10 0.663 1.368 2.82 1.03 0.389 -0.003 161012 1164686041.886 1.674e-7 9.385e-1 21.13 4.98 9.25 0.864 1.267 3.29 1.02 -0.997 0.003 170219 1171548267.039 1.744e-7 9.453e-1 21.09 8.66 4.10 1.313 1.270 2.81 1.14 0.850 -0.047 170423 1176984663.008 1.805e-7 9.506e-1 21.06 4.73 7.55 0.798 0.660 1.84 1.00 -0.050 -0.050 170412 1176047817.005 2.293e-7 9.781e-1 20.83 6.83 6.87 1.309 0.946 35.14 1.096 0.689 -0.037 151012A 1128666662.249 2.385e-7 9.812e-1 20.79 7.49 6.01 1.181 1.012 3.05 1.78 -0.297 0.014 161217 1165994201.408 2.823e-7 9.909e-1 20.64 8.96 5.91 1.771 1.498 88.56 1.57 0.823 -0.037 170630 1182874645.836 2.916e-7 9.922e-1 20.61 5.38 8.10 1.147 0.910 1.07 1.01 -0.038 -0.049 170720 1184625889.819 2.998e-7 9.932e-1 20.58 5.32 11.86 1.641 3.157 82.11 1.04 0.416 0.014 170705 1183279534.336 3.058e-7 9.939e-1 20.56 7.71 5.11 1.040 0.831 10.91 1.64 0.551 0.002 170208 1170585583.789 3.118e-7 9.945e-1 20.54 9.15 4.09 1.035 1.161 101.43 1.29 0.505 -0.040 170728 1185272107.732 3.538e-7 9.972e-1 20.42 9.96 4.69 1.615 0.584 72.42 1.18 -0.509 0.012 170729A 1185362055.597 3.599e-7 9.975e-1 20.41 4.37 8.13 0.725 1.009 169.34 3.79 0.998 0.898 170225 1172038323.230 3.607e-7 9.975e-1 20.41 7.77 5.43 1.076 1.483 3.35 1.35 -0.378 -0.049 Table 7.1. The significance and template parameters associated with all candidates with FARs estimated by the GstLAL-based inspiral pipeline to be below 1 per 30 days. The Poisson probability of observing 14 candidates from noise is 2.7σ, thus it is possible all of the marginal candidates are noise, however it is also possible one or more of them are marginal signals. As described in Table 2.3, the noise model used to produce the significance estimates is different than the noise model used to produce the published estimates, though this noise model has lower signal contamination. likelihood-ratio around. This is not expected to always happen for events quiet in one detector (e.g. see GW170818), and should not happen for signals loud in all detectors. The other point to note is that GW170608 is not included in Table 7.1 or the analysis described in this section. GW170608 occurred when only one detector, L1 had science quality data. The other detector was undergoing a routine procedure during which its data is typically not analyzed. After the existence of a chirp-like signal was discovered in L1, it was decided that data from H1 above 30 Hz was usable [82]. Thus data that would not normally be analyzed from H1 was analyzed to confirm the existence of GW170608. The significance estimates from this procedure do not produce statistically-meaningful numbers however. For example, the significance estimates could drastically change if all H1 data taken during these routine procedures were analyzed, instead of only times around the candidate. For this reason, the search-pipeline results for GW170608 are not included here. It is, however, used in the rate estimate and is included in

115 Fig. 7.1.

7.4 Comparison to Parameter Estimation Results

Parameter estimation was performed for all of the events from Sec. 7.3 that have a GW label. The O1 data used are the same as were used in Ref. [8] (where the GW151226 parameter estimation results in Sec. 3.5 were published). Although the data haven’t changed, the O1 events were re-analyzed because the parameter estimation analysis methods have improved since the original analyses [5]. The O2 data that are used are the same data that were used for the significance estimates (Sec. 7.3). The parameters for signals from binary black hole coalescences were estimated by averaging the posterior distributions from two different waveform approximations: IMRPhenomPv2 [42], which uses an effective precession model; and SEOB- NRv3 [40], which fully incorporates precession [5]. The parameters for GW170817 were estimated using several different waveforms, however the published mass and spin estimations were obtained using the IMRPhenomPv2NRT [42] waveform approximation, a frequency domain approximation that includes tidal effects [5]. The parameters were again estimated using a low-spin prior where the spins cannot be larger than 0.05 and a high-spin prior where the spins cannot be larger than 0.89 [5]. The chirp mass and effective spin parameter (χ) estimates from the GstLAL- based inspiral pipeline were within the 90% credible intervals for 6 of the 9 binary black hole signals: GW151012, GW170104, GW170729, GW170809, and GW170817. The χ point-estimate for GW150914 is within the 90% credible interval from parameter estimation, but M is not. The M and χ estimates from GW170814 and GW170823 did not fall within the 90% credible region estimated by parameter estimation. It is interesting to note that GW150914, GW170814, and GW170823 are among the shortest duration signals, along with GW170729 and GW170809. However, the 90% credible interval from GW170729 is very broad because of how quiet the signal is, so having parameters that fall within the credible interval is not as difficult. It is possible that there is a correlation between the duration of the signal in the analyzed frequency band and the pipeline’s ability to estimate its parameters, however with only 9 binary black hole signals it is too early to tell.

116 The M estimate for GW170817 is consistent with the 90% credible interval from both spin priors, though the component mass and χ estimates are only consistent with the parameter estimation results that use the high-spin prior. The parameter estimation results shown here were computed from the posterior samples publicly available through the Gravitational-Wave Open Science Center (gwosc) [2].

GstLAL Point Estimate Parameter Estimation Results +5.0 m1/M 46.89 38.9−3.2 +3.2 m2/M 32.33 33.5−4.9 +1.7 M/M 33.78 31.2−1.6 +4.0 M/M 79.22 72.2−3.4 +0.12 χ 0.0899 −0.01−0.13 Table 7.2. The signiﬁcance and template parameters associated with the most sig- niﬁcant GW150914 candidate found in the catalog analysis compared to the updated detector-frame median and 90% credible intervals from parameter estimation. As before (Table 2.4), most of the parameter point estimates are not within the 90% credible interval. Posterior samples hosted by the LIGO Open Science Center [2] were used to compute numbers shown here.

GstLAL Point Estimate Parameter Estimation Results +17.5 m1/M 31.73 28.1−6.4 +4.8 m2/M 13.65 16.6−6.0 +2.7 M/M 17.81 18.3−1.1 +12.5 M/M 45.38 44.9−3.7 +0.31 χ -0.0252 0.05−0.20 Table 7.3. The signiﬁcance and template parameters associated with the most sig- niﬁcant GW151012 candidate found in the catalog analysis compared to the updated detector-frame median and 90% credible intervals from parameter estimation. All of the point estimates are within the 90% credible intervals. Posterior samples hosted by the LIGO Open Science Center [2] were used to compute numbers shown here.

117 GstLAL Point Estimate Parameter Estimation Results +9.6 m1/M 20.19 15.0−3.5 +2.4 m2/M 6.57 8.4−2.8 +0.1 M/M 9.73 9.7−0.1 +6.8 M/M 27.76 23.4−1.2 +0.20 χ 0.3666 0.18−0.12 Table 7.4. The signiﬁcance and template parameters associated with the most sig- niﬁcant GW151226 candidate found in the catalog analysis compared to the updated detector-frame median and 90% credible intervals from parameter estimation. As before (Table 3.3), all of the point estimates are within the 90% credible intervals. Posterior samples hosted by the LIGO Open Science Center [2] were used to compute numbers shown here.

GstLAL Point Estimate Parameter Estimation Results +7.8 m1/M 55.19 36.9−6.2 +5.9 m2/M 15.90 24.0−5.7 +2.1 M/M 24.86 25.7−2.2 +4.8 M/M 71.09 61.1−4.4 +0.17 χ 0.1164 −0.04−0.21 Table 7.5. The signiﬁcance and template parameters associated with the most sig- niﬁcant GW170104 candidate found in the catalog analysis compared to the updated detector-frame median and 90% credible intervals from parameter estimation. As before (Table 4.3), the chirp mass and spin point estimates are within the 90% credible intervals, while the total mass and component mass estimates are not. Posterior samples hosted by the LIGO Open Science Center [2] were used to compute numbers shown here.

7.5 Estimated Rates

The Poisson-mixture model described in Sec. 2.6, which was used to estimate the number of gravitational wave signals from compact binaries in O1 and after the ﬁrst detection of O2, was updated to allow for diﬀerent categories of signals [5,83]. In this new framework, Eq. 2.3 becomes

118 GstLAL Point Estimate Parameter Estimation Results +20.5 m1/M 79.20 74.7−13.1 +12.6 m2/M 49.12 51.0−17.1 +7.4 M/M 53.99 53.3−9.2 +17.8 M/M 128.32 126.2−17.3 +0.21 χ 0.4138 0.37−0.25 Table 7.6. The significance and template parameters associated with the most significant GW170729 candidate found in the catalog analysis compared to the detector-frame median and 90% credible intervals from parameter estimation. All of the parameter point estimates are within the 90% credible intervals, although GW170729 is the qui- etest signal observed (Table 7.1), and as such has broader 90% confidence intervals than the other detections. Posterior samples hosted by the LIGO Open Science Center [2] were used to compute numbers shown here.

GstLAL Point Estimate Parameter Estimation Results +9.2 m1/M 47.12 42.0−6.8 +6.2 m2/M 23.61 28.6−6.4 +2.3 M/M 28.70 29.9−2.1 +5.5 M/M 70.73 70.9−4.6 +0.17 χ 0.0745 0.08−0.17 Table 7.7. The signiﬁcance and template parameters associated with the most signiﬁ- cant GW170809 candidate found in the catalog analysis compared to the detector-frame median and 90% credible intervals from parameter estimation. All of the parameter point estimates are within the 90% credible intervals. Posterior samples hosted by the LIGO Open Science Center [2] were used to compute numbers shown here.

~ ~ P Λ0, Λ1 {log Lj | j = 1,...,M} ∝ P Λ0, Λ1 " 3 # M " 3 # X Y X P (log Li| θi, S)P(θi| k, S) × exp − Λ1k Λ0 + Λ1k , (7.7) k=0 i=1 k=1 P (log Li| θi, N )P(θi| N ) where θi is the template bin that the ith candidate is from and the index k denotes the category, where 0 is noise and 1, 2, 3 correspond to the binary neutron star, neutron star black hole binary, and binary black hole categories. The signal

119 GstLAL Point Estimate Parameter Estimation Results +6.2 m1/M 59.14 34.3−3.2 +2.9 m2/M 15.23 28.3−4.5 +1.2 M/M 25.03 27.1−1.1 +3.2 M/M 74.37 62.8−2.6 +0.12 χ 0.2536 0.07−0.11 Table 7.8. The signiﬁcance and template parameters associated with the most sig- niﬁcant GW170814 candidate found in the catalog analysis compared to the updated detector-frame median and 90% credible intervals from parameter estimation. Unlike before (Table 5.3), none of the point estimates are within the 90% credible interval. There is no obvious reason this would happen, though it is possible there was a candidate with closer parameters that was slightly suppressed by noise in the Virgo detector (where the signal was barely loud enough to pass our SNR threshold), which is the expected reason GW170814 was recovered as a double-detector candidate instead of a triple-detector candidate. Posterior samples hosted by the LIGO Open Science Center [2] were used to compute numbers shown here.

GstLAL Point Estimate Parameter Estimation Results +8.7 m1/M 55.38 42.8−5.6 +5.3 m2/M 24.09 32.3−6.5 +2.4 M/M 31.27 32.2−2.3 +5.6 M/M 79.47 75.3−5.1 +0.18 χ -0.0826 −0.09−0.21 Table 7.9. The significance and template parameters associated with the most significant GW170818 candidate found in the catalog analysis compared to the detector-frame median and 90% credible intervals from parameter estimation. The chirp mass, total mass, and effective spin point estimates fall within the 90% credible regions. Posterior samples hosted by the LIGO Open Science Center [2] were used to compute numbers shown here.

category template bin weights, P(θi| k, S), were computed by using training sets of injections (i.e. not used to estimate rates) and histogramming the bins that the most signiﬁcant candidate for each found injection came from [83]. The noise template bin weight, P(θi| N ), was computed by histogramming the bins that the

120 GstLAL Point Estimate Parameter Estimation Results +12.2 m1/M 42.44 52.7−7.7 +8.1 m2/M 33.60 39.7−10.3 +4.8 M/M 32.83 39.4−4.5 +10.8 M/M 76.04 92.6−9.6 +0.20 χ -0.4579 0.08−0.23 Table 7.10. The signiﬁcance and template parameters associated with the most signiﬁ- cant GW170823 candidate found in the catalog analysis compared to the detector-frame median and 90% credible intervals from parameter estimation. Only the lighter component mass falls within the 90% credible region, though there is no obvious reason why. Posterior samples hosted by the LIGO Open Science Center [2] were used to compute numbers shown here.

GstLAL Point Estimate Parameter Estimation Results (Low Spin) Parameter Estimation Results (High Spin)

m1/M 1.78 1.38 − 1.60 1.38 − 1.86

m2/M 1.08 1.16 − 1.38 0.99 − 1.38 +0.0001 +0.0003 M/M 1.1977 1.1976−0.0001 1.1976−0.0002 +0.04 +0.15 M/M 2.85 2.76−0.01 2.80−0.04 χ 0.03 −0.01 − 0.02 −0.00 − 0.08 Table 7.11. The significance and template parameters associated with the most significant GW170817 candidate found in the catalog analysis compared to the detector-frame 90% credible intervals for the component masses and the median and 90% credible intervals for the other values from parameter estimation. As before (Table 6.3), the parameter estimation numbers were computed using two different spin priors: a low-spin prior motivated by the observed population of binary neutron star spins, and a high-spin prior. All of the parameter point estimates are within the high-spin prior 90% credible intervals. Posterior samples hosted by the LIGO Open Science Center [2] were used to compute numbers shown here. zerolag candidates came from. The latter process results in signal contamination, but the number of noise candidates is much, much greater than the number of signal candidates, so the effect is negligible [83]. In the end, Λ0 is marginalized ~ out to leave just P Λ {log Lj | j = 1,...,M} . In the limit of large M (M is the number of candidates) and when the candidate list consists overwhelmingly of noise candidates, the posterior for Λ0 can be reasonably approximated as a Dirac delta function around M, making the marginalization trivial [83]. The injection sets used to estimate hVTi were analyzed using the method I de-

121 veloped to reduce the cost of injection analyses (Sec. 6.6). The average space-time volume in which the detectors are sensitive to binary black hole mergers was again estimated using a uniform-in-log-mass distribution and a power-law distribution for the heavier black hole. One difference however is that the component mass upper bound was reduced from 100M to 50M [5]. An analysis of the binary black hole coalescences observed in O1 and O2 showed that the upper mass cutoff for the power-law distribution would be around 50M [84], though there was support in the literature for this cutoff before that analysis [5]. The median and 90% credible interval for the uniform-in-log-mass distribution from the GstLAL-based +13.9 −3 −1 +33 −3 −1 inspiral pipeline is 18.1−8.7 Gpc yr , compared to 32−20 Gpc yr from before +44 −3 −1 (Sec 4.5); the rate estimate using the power-law distribution is 56−27 Gpc yr , +110 −3 −1 compared to 103−63 Gpc yr from before. The average space-time volume that the detectors are sensitive to binary neutron stars from was also estimated using two different component mass distributions: uniform between 1 M and 2 M to obtain an update of the number published in the GW170817 detection paper [11]; and a Gaussian distribution with mean 1.33 M and a standard deviation of 0.09 M , allowing masses between 1 M and 3 M . The median and 90% credible interval estimated from the GstLAL-based inspiral pipeline using the uniform-in-component-mass distribution +1609 −3 −1 +2920 −3 −1 was 662−565 Gpc yr , compared to 1420−1150 Gpc yr before; the rate esti- +2220 −3 −1 mated for the uncorrelated Gaussian distributions model was 920−790 Gpc yr . There were no confident neutron star black hole detections during O1 or O2 (though there are marginal candidates in Table 7.1), meaning that we can only place an upper-limit on the merger rate. This is done the same way that the binary neutron star merger rate was estimated in Ref. [11] (Sec. 6.6). However, this time we use the likelihood of observing zero candidates instead of observing one, changing Eq. 6.2 to

Z P(R| d) ∝ dhVTi P(R| hVTi)P(hVTi) e−RhVTi, (7.8) where a uniform prior was used for the rate. The rate posterior was estimated via a Markov-chain Monte Carlo algorithm for three diﬀerent black hole masses, each paired with a neutron star of mass 1.4 M . For systems where the black hole mass is 5 M and the spins are aligned, the 90% upper limit computed using the

122 GstLAL-based inspiral pipeline’s estimate of hVTi is 537 Gpc−3 yr−1, the upper limit for the same mass but with an isotropic spin distribution is 605 Gpc−3 yr−1; the 90% upper limit for systems with a black hole mass of 10 M and aligned spins is 341 Gpc−3 yr−1 while the upper limit for the isotropic spin distribution is −3 −1 475 Gpc yr ; the 90% upper limit for systems with a black hole mass of 30 M and aligned spins is 218 Gpc−3 yr−1; while the upper limit for the isotropic spin distribution is 402 Gpc−3 yr−1.

123 Chapter 8 | Low-Latency Alerts in O3

8.1 Introduction

LIGO’s third observing run, O3, began April 1, 2019, UTC. The sensitivity of all of the detectors has increased relative to where they were at at the end of O2, and as of Jun 2, 2019, 11 binary black hole merger candidates, 4 binary neutron star merger candidates, and 1 potential neutron star black hole binary merger candidate have been identiﬁed. Two of the binary neutron star merger candidates have been retracted after non-stationary noise was noticed near the time of the candidates.

8.2 Methods

8.2.1 Bank

The template bank the low-latency analysis is using at the time of writing is the same bank that was used offline at the end of O2, described in Sec. 5.2.1.2 and more in-depth in Ref. [3]. Originally, the low-latency analysis ran with a different binning than offline analysis ran with. Instead of using two different binning techniques on low-mass and high-mass templates (Sec. 5.2.1.2 and Ref. [9]), the different regions of the bank (BNS, NSBH, BBH, IMBH) were binned only amongst themselves. The binning algorithm was the original M and χ binning (Ref. [4]). Only binning templates with other templates in the same signal category (e.g. only bin IMBH templates with other IMBH templates) is a new method that was discovered to

124 mitigate the problem of waveforms with diﬀerent noise properties getting grouped together (Sec. 5.2). About a week into O3, the low-latency analysis was replaced with a new analysis, run by the same code but using the binning used for the catalog results (Ref. [5] and Chapter 7). The switch was made so that the weights computed for the multi- component Poisson-mixture model (Ref. [5,83] and Sec. 7.5) in O2 could be used to estimate the probability of each candidate being an astrophysical candidate from one of the signal categories.

8.2.2 Likelihood-ratio

During O1 and O2, signals were considered to be equally likely in all templates, i.e. ¯ P θ S from Eq. 1.140 was uniform. Between O2 and O3, machinery was added to the GstLAL-based inspiral pipeline to force users to choose a signal model. For alerts in O3 so far, a basic analytical signal model that only incorporates the chirp mass has been used, shown in Fig. 8.1. The binary neutron star region is modeled as a Gaussian distribution, with a mean chirp mass of M = 1.2 M and

σ = .25 M . The binary black hole region is a power-law distribution with an index of −1.5. The two functions were smoothed to meet smoothly in-between. The intention of this simple mass model is to improve our sensitivity to marginal signals similar to those already observed without negatively aﬀecting our ability to observe loud signals that are unlike those previously seen.

8.2.3 Itacac

The trigger generator, which took SNR time-series as input and produced triggers as output, used during O1 and O2 was named itac, which stands for inspiral trigger and autocorrelation chi-squared1. Each instance of itac in the GStreamer pipeline could only ingest an SNR time-series from one detector, which meant that information about SNR peaks below the trigger threshold (usually 4.0) was lost before the coincidence stage (see Fig. 8.2(a)). As a result, only information from the above-threshold detectors was automatically uploaded to GraceDB [10] and used by BAYESTAR [7] to estimate skymaps. Adding information from a third

1The autocorrelation signal-based veto, ξ2, is internally referred to as χ2. ξ2 is used in publications to avoid potential confusion due to our χ2 test-statistic not following a χ2 distribution.

125 (a) (b)

Figure 8.1. The analytical signal model being used in low-latency alerts in O3 at the time of writing. The binary neutron star region is assumed to follow a Gaussian distribution in chirp mass, with µ = 1.2 M and σ = .25 M , while the binary black hole region is assumed to follow a power-law distribution in chirp mass with an index of −1.5. The relative FARs in these plots are only to show how their FAR would be aﬀected relative to other templates in the plot. The left plot is zoomed in on low masses, while the right plot shows the entire template bank. The plots were both normalized individually, thus the normalizations of the color bars are not expected to be the same. detector, even when the signal is quiet in that detector, can decrease the area of the 90% credible region in the skymap by over an order of magnitude [11]. Originally, itac ingested SNR time-series from only one detector because syn- chronizing multiple time-series streams requires a large amount of extra bookkeeping. The decision was made to change this after O2; fortunately the newest release of GStreamer at the time, 1.14, contained a new base class that performs a lot of the bookkeeping for the user. Using this new class, I wrote a new GStreamer element based on itac named itacac, standing for inspiral trigger, autocorrelation chi-squared, and coincidence. Despite the name, at the moment itacac does not perform coincidence (see Fig. 8.2(b)). However, for each above-threshold SNR peak itacac saves SNR time-series for each detector centered on the time of the peak. When a low-latency candidate passes the upload threshold, sub-threshold triggers are generated from the surrounding time series if any of the detectors online at the time of the candidate did not have an SNR peak above threshold. So far, several of the alerts issued in O3 contain sub-threshold triggers to help with localization, though the SNRs are not currently public so the candidates containing sub-threshold triggers cannot be identiﬁed at the time of writing. I plan to

126 (a) (b)

Figure 8.2. A high-level depiction of the workﬂow in the GstLAL-based inspiral pipeline. Before O3, SNR time-series were converted to triggers one detector at a time by an in-house GStreamer element named itac, meaning that sub-threshold SNR information is lost before uploading to GraceDB. Now the SNR time-series from all detectors are fed into a new in-house GStreamer element named itacac, which converts the time-series from diﬀerent detectors to triggers at the same time. This means that if one (or more) detectors have an SNR below the trigger threshold, SNR from that detector can still be saved and uploaded to GraceDB [10] to be used in rapid localization. add coincidence to itacac in the future.

8.3 Public Alerts To Date

There have been 16 public alerts as of June 2, 2019, although there was a 17th that was sent in error and immediately retracted. The masses and estimated signiﬁcances are not public information as of this moment, but each candidate has a GCN circular and a GraceDB page. The candidates, and their most likely classiﬁcation, are listed below, sorted by date. Note that there was a naming convention change before O3, low-latency candidates are now labeled by a number that begins with S, then has a 6 digit date, and followed by one or more letters (how ever many are needed to make it unique).

• S190408an, likely binary black hole merger [85]

127 • S190412m, likely binary black hole merger [86]

• S190421ar, likely binary black hole merger [87]

• S190425z, likely binary neutron star merger [88]

• S190426c, possible neutron star black hole binary merger [89]

• S190503bf, likely binary black hole merger [90]

• S190510g, likely binary neutron star merger [91]

• S190512at, likely binary black hole merger [92]

• S190513bm, likely binary black hole merger [93]

• S190517h, likely binary black hole merger [94]

• S190518bb, binary neutron star merger candidate, retracted [95]

• S190519bj, likely binary black hole merger [96]

• S190521g, likely binary black hole merger [97]

• S190521r, likely binary black hole merger [98]

• S190524q, binary neutron star merger candidate, retracted [99]

• S190602aq, likely binary black hole merger [100]

The majority of candidates observed so far are likely binary black hole mergers, though there is one possible neutron star black hole merger candidate and two likely binary neutron star mergers. There have been two retracted alerts, both of which were binary neutron star candidates. It is still possible the retracted candidates are real, but we will not be able to say either way until an oﬄine analysis of O3 is done using improved calibration.

128 8.4 Current State of the Field

The field of gravitational-wave astronomy is in a transitionary period. When I began graduate school, many in the scientific community were skeptical that gravitational waves could be detected if they even existed. Now we’ve observed enough that we’re reasonably confident in a 1.40σ candidate (GW170729). Additionally, all significant gravitational-wave detections are immediately made public, a stark change from the beginning of O1 when GW150914 was observed in September but not announced until the following February. This has resulted in a push to analyze data in low-latency that did not exist before O1. The GstLAL-based inspiral pipeline was the only low-latency compact binary analysis developed by members of the LSC that was running during O1, now there are three. We have gone from having to convince our colleagues to search for binary black hole mergers in low-latency to competing with them to find the signals first. I believe the most likely next short-timescale-steps in the field of compact binary gravitational waves are, in no specific order: detect a 5σ neutron-star black hole binary merger; detect enough binary black hole mergers to distinguish between different formation models; detect enough binary neutron star mergers to begin testing if the population observed via gravitational waves is the same as the population observed electromagnetically; and detect a 5σ intermediate mass black hole binary (component masses & 100 M ). The era of public alerts means that if there are observable multi-messenger counterparts to any of gravitational-wave signals believed to come from mergers involving at least one neutron star, then we are likely to see it. The conventional wisdom of the field at the time of writing is that we are unlikely to observe another multi-messenger event as magnificent as GW170817 in the near future, though the conventional wisdom in 2015 before O1 began was that we were unlikely to observe a multi-messenger event with this generation of gravitational-wave detectors. The public alert era also opens the field up to more exploratory studies, such as following up binary black hole mergers in low-latency to look for unpredicted multi- messenger counterparts. As the field is young, there are many similar opportunities to look for new physics, and any scientist capable of analyzing LIGO data has the potential to make an unexpected but exciting discovery.

129 8.5 Conclusion

The purpose of my dissertation research was to detect gravitational waves from merging compact binaries as quickly as possible in order to enable multi-messenger astronomy. On August 17, 2017, I was the first person to see that we had detected gravitational waves from a merging binary neutron star, which led to the first extra-galactic multi-messenger detection in 30 years. In addition, I’ve contributed to every gravitational wave detection to date. I plan to continue my involvement in discovering gravitational-waves from merging compact binaries as a postdoctoral scholar. The work presented in this dissertation is a summary of the first few years of gravitational-wave astronomy, from the first detection to the first public detections. Fifty years ago we had no observational evidence that compact binaries form close enough to merge within the lifetime of the universe, now we’re seeing roughly one binary black hole merger per week. Furthermore, detections that were ground- breaking in 2015 are now mundane. The era of gravitational-wave astronomy is here, and it’s clear that we have only scratched the surface of what we can learn from these cosmic ripples.

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140 Vita Cody Messick

Education The Pennsylvania State University, Ph.D., Physics, 2013-2019 University of Washington, Bachelor of Science, Comprehensive Physics with De- partmental Honors, 2011-2013 Clark College, 2008-2010

Selected Publications Abbott, B. P., et al. “GWTC-1: A Gravitational-Wave Transient Catalog of Com- pact Binary Mergers Observed by LIGO and Virgo during the First and Second Observing Runs.” Abbott, B. P., et al. “GW170817: observation of gravitational waves from a binary neutron star inspiral.” Abbott, B. P., et al. “Gravitational waves and gamma-rays from a binary neutron star merger: GW170817 and GRB 170817A.” Abbott, B. P., et al. “Multi-messenger observations of a binary neutron star merger.” Abbott, B. P., et al. “GW170814: A three-detector observation of gravitational waves from a binary black hole coalescence.” Messick, Cody, et al. “Analysis framework for the prompt discovery of compact binary mergers in gravitational-wave data.” Abbott, B. P., et al. “GW151226: Observation of gravitational waves from a 22- solar-mass binary black hole coalescence.” Abbott, B. P., et al. “GW150914: First results from the search for binary black hole coalescence with Advanced LIGO.”

Selected Awards and Fellowships Frymoyer Honors Fellowship (Apr 2019) Alumni Association Dissertation Award (Mar 2019) Peter Eklund Memorial Lectureship Award (Feb 2019) Peter Eklund Memorial Lectureship Award Honorable Mention (Apr 2018) Frymoyer Honors Fellowship (Nov 2017) Clark College Foundation Rising Star Alumni Award (Oct 2017) Academic Computing Fellowship (Aug 2017) Frymoyer Honors Fellowship (Nov 2016) Special Breakthrough Prize in Fundamental Physics (May 2016) David C. Duncan Graduate Fellowship in Physics (Nov 2015) David C. Duncan Graduate Fellowship in Physics (Nov 2014) Verne M. Willaman Distinguished Graduate Fellowship in Science (Aug 2013)