GW150914: First results from the search for binary black hole coalescence with Advanced LIGO
On September 14, 2015 at 09:50:45 UTC the two detectors of the Laser Interferometer Gravitational-wave Observatory (LIGO) simultaneously observed the binary black hole merger GW150914. We report the results of a matched-filter search using relativistic models of compact-object binaries that recovered GW150914 as the most significant event during the coincident observations between the two LIGO detectors from September 12 to October 20, 2015. GW150914 was observed with a matched filter signal-to-noise ratio of and a false alarm rate estimated to be less than 1 event per years, equivalent to a significance greater than .
On September 14, 2015 at 09:50:45 UTC the LIGO Hanford, WA, and Livingston, LA, observatories detected a signal from the binary black hole merger GW150914 Abbott et al. (2016a). The initial detection of the event was made by low-latency searches for generic gravitational-wave transients Abbott et al. (2016b). We report the results of a matched-filter search using relativistic models of compact binary coalescence waveforms that recovered GW150914 as the most significant event during the coincident observations between the two LIGO detectors from September 12 to October 20, 2015. This is a subset of the data from Advanced LIGO’s first observational period that ended on January 12, 2016.
The binary coalescence search targets gravitational-wave emission from compact-object binaries with individual masses from 1 to 99, total mass less than 100 and dimensionless spins up to 0.99. The search was performed using two independently implemented analyses, referred to as PyCBC Dal Canton et al. (2014); Usman et al. (2015); Nitz et al. (2016) and GstLAL Cannon et al. (2012); Privitera et al. (2014); Messick et al. (2016). These analyses use a common set of template waveforms Taracchini et al. (2014); Pürrer (2016); Capano et al. (2016), but differ in their implementations of matched filtering Allen et al. (2012); Cannon et al. (2010a), their use of detector data-quality information Abbott et al. (2016c), the techniques used to mitigate the effect of non-Gaussian noise transients in the detector Allen (2005); Cannon et al. (2012), and the methods for estimating the noise background of the search Usman et al. (2015); Cannon et al. (2015).
GW150914 was observed in both LIGO detectors Abbott et al. (2016d) with a time-of-arrival difference of 7 ms, which is less than the 10 ms inter-site propagation time, and a combined matched-filter signal to noise ratio (SNR) of . The search reported a false alarm rate estimated to be less than 1 event per years, equivalent to a significance greater than . The basic features of the GW150914 signal point to it being produced by the coalescence of two black holes Abbott et al. (2016a). The best-fit template parameters from the search are consistent with detailed parameter estimation that identifies GW150914 as a near-equal mass black hole binary system with source-frame masses and at the credible level Abbott et al. (2016e).
The second most significant candidate event in the observation period (referred to as LVT151012) was reported on October 12, 2015 at 09:54:43 UTC with a combined matched-filter SNR of . The search reported a false alarm rate of 1 per 2.3 years and a corresponding false alarm probability of 0.02 for this candidate event. Detector characterization studies have not identified an instrumental or environmental artifact as causing this candidate event Abbott et al. (2016c). However, its false alarm probability is not sufficiently low to confidently claim this candidate event as a signal Gonzalez et al. (2012e). Detailed waveform analysis of this candidate event indicates that it is also a binary black hole merger with source frame masses and , if it is of astrophysical origin.
This paper is organized as follows: Sec. II gives an overview of the compact binary coalescence search and the methods used. Sec. III and Sec. IV describe the construction and tuning of the two independently implemented analyses used in the search. Sec. V presents the results of the search, and follow-up of the two most significant candidate events, GW150914 and LVT151012.
Ii Search Description
The binary coalescence search Thorne (1987); Sathyaprakash and Dhurandhar (1991); Cutler et al. (1993); Finn (1992); Finn and Chernoff (1993); Dhurandhar and Sathyaprakash (1994); Balasubramanian et al. (1996); Flanagan and Hughes (1998) reported here targets gravitational waves from binary neutron stars, binary black holes, and neutron star–black hole binaries, using matched filtering Wainstein and Zubakov (1962) with waveforms predicted by general relativity. Both the PyCBC and GstLAL analyses correlate the detector data with template waveforms that model the expected signal. The analyses identify candidate events that are detected at both observatories consistent with the ms inter-site propagation time. Events are assigned a detection-statistic value that ranks their likelihood of being a gravitational-wave signal. This detection statistic is compared to the estimated detector noise background to determine the probability that a candidate event is due to detector noise.
We report on a search using coincident observations between the two Advanced LIGO detectors Aasi et al. (2015) in Hanford, WA (H1) and in Livingston, LA (L1) from September 12 to October 20, 2015. During these days, the detectors were in coincident operation for a total of 18.4 days. Unstable instrumental operation and hardware failures affected 20.7 hours of these coincident observations. These data are discarded and the remaining 17.5 days are used as input to the analyses Abbott et al. (2016c). The analyses reduce this time further by imposing a minimum length over which the detectors must be operating stably; this is different between the two analysis (2064 s for PyCBC and 512 s for GstLAL), as described in Sec. III and Sec. IV. After applying this cut, the PyCBC analysis searched of coincident data and the GstLAL analysis searched days of coincident data. To prevent bias in the results, the configuration and tuning of the analyses were determined using data taken prior to September 12, 2015.
A gravitational-wave signal incident on an interferometer alters its arm lengths by and , such that their measured difference is , where is the gravitational-wave metric perturbation projected onto the detector, and is the unperturbed arm length Abramovici et al. (1992). The strain is calibrated by measuring the detector’s response to test mass motion induced by photon pressure from a modulated calibration laser beam Abbott et al. (2016f). Changes in the detector’s thermal and alignment state cause small, time-dependent systematic errors in the calibration Abbott et al. (2016f). The calibration used for this search does not include these time-dependent factors. Appendix A demonstrates that neglecting the time-dependent calibration factors does not affect the result of this search.
The gravitational waveform depends on the chirp mass of the binary, Peters and Mathews (1963); Peters (1964), the symmetric mass ratio Blanchet et al. (1995), and the angular momentum of the compact objects Kidder et al. (1993); Kidder (1995) (the compact object’s dimensionless spin), where is the angular momentum of the compact objects. The effect of spin on the waveform depends also on the ratio between the component objects’ masses Blanchet (2014). Parameters which affect the overall amplitude and phase of the signal as observed in the detector are maximized over in the matched-filter search, but can be recovered through full parameter estimation analysis Abbott et al. (2016e). The search parameter space is therefore defined by the limits placed on the compact objects’ masses and spins. The minimum component masses of the search are determined by the lowest expected neutron star mass, which we assume to be Miller and Miller (2014). There is no known maximum black hole mass Belczynski et al. (2014), however we limit this search to binaries with a total mass less than . The LIGO detectors are sensitive to higher mass binaries, however; the results of searches for binaries that lie outside this search space will be reported in future publications.
The limit on the spins of the compact objects are informed by radio and X-ray observations of compact-object binaries. The shortest observed pulsar period in a double neutron star system is ms Burgay et al. (2003), corresponding to a spin of . Observations of X-ray binaries indicate that astrophysical black holes may have near extremal spins McClintock et al. (2013). In constructing the search, we assume that compact objects with masses less than are neutron stars and we limit the magnitude of the component object’s spin to . For higher masses, the spin magnitude is limited to with the upper limit set by our ability to generate valid template waveforms at high spins Taracchini et al. (2014). At current detector sensitivity, limiting spins to for does not reduce the search sensitivity for sources containing neutron stars with spins up to , the spin of the fastest-spinning millisecond pulsar Lorimer (2008). Figure 1 shows the boundaries of the search parameter space in the component-mass plane, with the boundaries on the mass-dependent spin limits indicated.
Since the parameters of signals are not known in advance, each detector’s output is filtered against a discrete bank of templates that span the search target space Sathyaprakash and Dhurandhar (1991); Owen (1996); Owen and Sathyaprakash (1999); Babak et al. (2006); Cokelaer (2007). The placement of templates depends on the shape of the power spectrum of the detector noise. Both analyses use a low-frequency cutoff of Hz for the search. The average noise power spectral density of the LIGO detectors was measured over the period September 12 to September 26, 2015. The harmonic mean of these noise spectra from the two detectors was used to place a single template bank that was used for the duration of the search Keppel (2013); Usman et al. (2015). The templates are placed using a combination of geometric and stochastic methods Harry et al. (2009); Brown et al. (2012); Privitera et al. (2014); Capano et al. (2016) such that the loss in matched-filter SNR caused by its discrete nature is %. Approximately 250,000 template waveforms are used to cover this parameter space, as shown in Fig. 1.
The performance of the template bank is measured by the fitting factor Apostolatos (1996); this is the fraction of the maximum signal-to-noise ratio that can be recovered by the template bank for a signal that lies within the region covered by the bank. The fitting factor is measured numerically by simulating a signal and determining the maximum recovered matched-filter SNR over the template bank. Figure 2 shows the resulting distribution of fitting factors obtained for the template bank over the observation period. The loss in matched-filter SNR is less than for more than % of the simulated signals.
The template bank assumes that the spins of the two compact objects are aligned with the orbital angular momentum. The resulting templates can nonetheless effectively recover systems with misaligned spins in the parameter-space region of GW150914. To measure the effect of neglecting precession in the template waveforms, we compute the effective fitting factor which weights the fraction of the matched-filter SNR recovered by the amplitude of the signal Buonanno et al. (2003). When a signal with a poor orientation is projected onto the detector, the amplitude of the signal may be too small to detect even if there was no mismatch between the signal and the template; the weighting in the effective fitting accounts for this. Figure 3 shows the effective fitting factor for simulated signals from a population of simulated precessing binary black holes that are uniform in co-moving volume Pan et al. (2014); Harry et al. (2016). The effective fitting factor is lowest at high mass ratios and low total mass, where the effects of precession are more pronounced. In the region close to the parameters of GW150914 the aligned-spin template bank is sensitive to a large fraction of precessing signals Harry et al. (2016).
In addition to possible gravitational-wave signals, the detector strain contains a stationary noise background that primarily arises from photon shot noise at high frequencies and seismic noise at low frequencies. In the mid-frequency range, detector commissioning has not yet reached the point where test mass thermal noise dominates, and the noise at mid frequencies is poorly understood Abbott et al. (2016d, c); Martynov et al. (2016). The detector strain data also exhibits non-stationarity and non-Gaussian noise transients that arise from a variety of instrumental or environmental mechanisms. The measured strain is the sum of possible gravitational-wave signals and the different types of detector noise .
To monitor environmental disturbances and their influence on the detectors, each observatory site is equipped with an array of sensors Effler et al. (2015). Auxiliary instrumental channels also record the interferometer’s operating point and the state of the detector’s control systems. Many noise transients have distinct signatures, visible in environmental or auxiliary data channels that are not sensitive to gravitational waves. When a noise source with known physical coupling between these channels and the detector strain data is active, a data-quality veto is created that is used to exclude these data from the search Abbott et al. (2016c). In the GstLAL analysis, time intervals flagged by data quality vetoes are removed prior to the filtering. In the PyCBC analysis, these data quality vetoes are applied after filtering. A total of 2 hours is removed from the analysis by data quality vetoes. Despite these detector characterization investigations, the data still contains non-stationary and non-Gaussian noise which can affect the astrophysical sensitivity of the search. Both analyses implement methods to identify loud, short-duration noise transients and remove them from the strain data before filtering.
The PyCBC and GstLAL analyses calculate the matched-filter SNR for each template and each detector’s data Allen et al. (2012); Cannon et al. (2010b). In the PyCBC analysis, sources with total mass less than 4 are modeled by computing the inspiral waveform accurate to third-and-a-half post-Newtonian order Blanchet et al. (1995); Droz et al. (1999); Blanchet et al. (2004). To model systems with total mass larger than 4, we use templates based on the effective-one-body (EOB) formalism Buonanno and Damour (2000), which combines results from the Post-Newtonian approach Blanchet et al. (1995, 2004) with results from black hole perturbation theory and numerical relativity Taracchini et al. (2014); Pürrer (2014) to model the complete inspiral, merger and ringdown waveform. The waveform models used assume that the spins of the merging objects are aligned with the orbital angular momentum. The GstLAL analysis uses the same waveform families, but the boundary between Post-Newtonian and EOB models is set at . Both analyses identify maxima of the matched-filter SNR (triggers) over the signal time of arrival.
To suppress large SNR values caused by non-Gaussian detector noise, the two analyses calculate additional tests to quantify the agreement between the data and the template. The PyCBC analysis calculates a chi-squared statistic to test whether the data in several different frequency bands are consistent with the matching template Allen (2005). The value of the chi-squared statistic is used to compute a re-weighted SNR for each maxima. The GstLAL analysis computes a goodness-of-fit between the measured and expected SNR time series for each trigger. The matched-filter SNR and goodness-of-fit values for each trigger are used as parameters in the GstLAL ranking statistic.
Both analyses enforce coincidence between detectors by selecting trigger pairs that occur within a ms window and come from the same template. The ms window is determined by the ms inter-site propagation time plus ms for uncertainty in arrival time of weak signals. The PyCBC analyses discards any triggers that occur during the time of data-quality vetoes prior to computing coincidence. The remaining coincident events are ranked based on the quadrature sum of the re-weighted SNR from both detectors Usman et al. (2015). The GstLAL analysis ranks coincident events using a likelihood ratio that quantifies the probability that a particular set of concident trigger parameters is due to a signal versus the probability of obtaining the same set of parameters from noise Cannon et al. (2012).
The significance of a candidate event is determined by the search background. This is the rate at which detector noise produces events with a detection-statistic value equal to or higher than the candidate event (the false alarm rate). Estimating this background is challenging for two reasons: the detector noise is non-stationary and non-Gaussian, so its properties must be empirically determined; and it is not possible to shield the detector from gravitational waves to directly measure a signal-free background. The specific procedure used to estimate the background is different for the two analyses.
To measure the significance of candidate events, the PyCBC analysis artificially shifts the timestamps of one detector’s triggers by an offset that is large compared to the inter-site propagation time, and a new set of coincident events is produced based on this time-shifted data set. For instrumental noise that is uncorrelated between detectors this is an effective way to estimate the background. To account for the search background noise varying across the target signal space, candidate and background events are divided into three search classes based on template length. To account for having searched multiple classes, the measured significance is decreased by a trials factor equal to the number of classes Lyons (2008).
The GstLAL analysis measures the noise background using the distribution of triggers that are not coincident in time. To account for the search background noise varying across the target signal space, the analysis divides the template bank into 248 bins. Signals are assumed to be equally likely across all bins and it is assumed that noise triggers are equally likely to produce a given SNR and goodness-of-fit value in any of the templates within a single bin. The estimated probability density function for the likelihood statistic is marginalized over the template bins and used to compute the probability of obtaining a noise event with a likelihood value larger than that of a candidate event.
The result of the independent analyses are two separate lists of candidate events, with each candidate event assigned a false alarm probability and false alarm rate. These quantities are used to determine if a gravitational-wave signal is present in the search. Simulated signals are added to the input strain data to validate the analyses, as described in Appendix B.
Iii PyCBC Analysis
The PyCBC analysis Dal Canton et al. (2014); Usman et al. (2015); Nitz et al. (2016) uses fundamentally the same methods Brown et al. (2004); Allen et al. (2012); Allen (2005); Brown (2005); Babak et al. (2013); Brown et al. (2006); Deelman et al. (2005, 2015); Thain et al. (2005); Couvares et al. (2006); Jones et al. (01); Walt et al. (2011); Hunter (2007) as those used to search for gravitational waves from compact binaries in the initial LIGO and Virgo detector era Abbott et al. (2004, 2005a, 2005b, 2006, 2008a, 2008b, 2009a, 2009b); Abadie et al. (2010, 2012); Aasi et al. (2013, 2014), with the improvements described in Refs. Dal Canton et al. (2014); Usman et al. (2015). In this Section, we describe the configuration and tuning of the PyCBC analysis used in this search. To prevent bias in the search result, the configuration of the analysis was determined using data taken prior to the observation period searched. When GW150914 was discovered by the low-latency transient searches Abbott et al. (2016a), all tuning of the PyCBC analysis was frozen to ensure that the reported false alarm probabilities are unbiased. No information from the low-latency transient search is used in this analysis.
Of the 17.5 days of data that are used as input to the analysis, the PyCBC analysis discards times for which either of the LIGO detectors is in their observation state for less than s; shorter intervals are considered to be unstable detector operation by this analysis and are removed from the observation time. After discarding time removed by data-quality vetoes and periods when detector operation is considered unstable the observation time remaining is .
For each template and for the strain data from a single detector , the analysis calculates the square of the matched-filter SNR defined by Allen et al. (2012)
where the correlation is defined by
where is the Fourier transform of the time domain quantity given by
The quantity is the one-sided average power spectral density of the detector noise, which is re-calculated every 2048 s (in contrast to the fixed spectrum used in template bank construction). Calculation of the matched-filter SNR in the frequency domain allows the use of the computationally efficient Fast Fourier Transform int (2015); Frigo and Johnson (2005). The square of the matched-filter SNR in Eq. (1) is normalized by
so that its mean value is , if contains only stationary noise Cutler and Flanagan (1994).
Non-Gaussian noise transients in the detector can produce extended periods of elevated matched-filter SNR that increase the search background Usman et al. (2015). To mitigate this, a time-frequency excess power (burst) search Robinet (2015) is used to identify high-amplitude, short-duration transients that are not flagged by data-quality vetoes. If the burst search generates a trigger with a burst SNR exceeding , the PyCBC analysis vetoes these data by zeroing out s of centered on the time of the trigger. The data is smoothly rolled off using a Tukey window during the s before and after the vetoed data. The threshold of is chosen to be significantly higher than the burst SNR obtained from plausible binary signals. For comparison, the burst SNR of GW150914 in the excess power search is . A total of burst-transient vetoes are produced in the two detectors, resulting in s of data removed from the search. A time-frequency spectrogram of the data at the time of each burst-transient veto was inspected to ensure that none of these windows contained the signature of an extremely loud binary coalescence.
The analysis places a threshold of on the single-detector matched-filter SNR and identifies maxima of with respect to the time of arrival of the signal. For each maximum we calculate a chi-squared statistic to determine whether the data in several different frequency bands are consistent with the matching template Allen (2005). Given a specific number of frequency bands , the value of the reduced is given by
where is the sub-template corresponding to the -th frequency band. Values of near unity indicate that the signal is consistent with a coalescence. To suppress triggers from noise transients with large matched-filter SNR, is re-weighted by Abadie et al. (2012); Babak et al. (2013)
Triggers that have a re-weighted SNR or that occur during times subject to data-quality vetoes are discarded.
The template waveforms span a wide region of time-frequency parameter space and the susceptibility of the analysis to a particular type of noise transient can vary across the search space. This is demonstrated in Fig. d which shows the cumulative number of noise triggers as a function of re-weighted SNR for Advanced LIGO engineering run data taken between September 2 and September 9, 2015. The response of the template bank to noise transients is well characterized by the gravitational-wave frequency at the template’s peak amplitude, . Waveforms with a lower peak frequency have less cycles in the detector’s most sensitive frequency band from – Hz Abbott et al. (2016d); Martynov et al. (2016), and so are less easily distinguished from noise transients by the re-weighted SNR.
The number of bins in the test is a tunable parameter in the analysis Usman et al. (2015). Previous searches used a fixed number of bins Babak et al. (2005) with the most recent Initial LIGO and Virgo searches using bins for all templates Abadie et al. (2012); Aasi et al. (2013). Investigations on data from LIGO’s sixth science run Nitz (2015); Aasi et al. (2013) showed that better noise rejection is achieved with a template-dependent number of bins. The left two panels of Fig. d show the cumulative number of noise triggers with bins used in the test. Empirically, we find that choosing the number of bins according to
gives better suppression of noise transients in Advanced LIGO data, as shown in the right panels of Fig. d.
The PyCBC analysis enforces signal coincidence between detectors by selecting trigger pairs that occur within a ms window and come from the same template. We rank coincident events based on the quadrature sum of the from both detectors Usman et al. (2015). The final step of the analysis is to cluster the coincident events, by selecting those with the largest value of in each time window of s. Any other events in the same time window are discarded. This ensures that a loud signal or transient noise artifact gives rise to at most one candidate event Usman et al. (2015).
The significance of a candidate event is determined by the rate at which detector noise produces events with a detection-statistic value equal to or higher than that of the candidate event. To measure this, the analysis creates a “background data set” by artificially shifting the timestamps of one detector’s triggers by many multiples of s and computing a new set of coincident events. Since the time offset used is always larger than the time-coincidence window, coincident signals do not contribute to this background. Under the assumption that noise is not correlated between the detectors Abbott et al. (2016c), this method provides an unbiased estimate of the noise background of the analysis.
To account for the noise background varying across the target signal space, candidate and background events are divided into different search classes based on template length. Based on empirical tuning using Advanced LIGO engineering run data taken between September 2 and September 9, 2015, we divide the template space into three classes according to: (i) ; (ii) and Hz; (iii) and Hz. The significance of candidate events is measured against the background from the same class. For each candidate event, we compute the false alarm probability . This is the probability of finding one or more noise background events in the observation time with a detection-statistic value above that of the candidate event, given by Usman et al. (2015); Capano et al. (2016)
where is the observation time of the search, is the background time, and is the number of noise background triggers above the candidate event’s re-weighted SNR .
Eq. (8) is derived assuming Poisson statistics for the counts of time-shifted background events, and for the count of coincident noise events in the search Usman et al. (2015); Capano et al. (2016). This assumption requires that different time-shifted analyses (i.e. with different relative shifts between detectors) give independent realizations of a counting experiment for noise background events. We expect different time shifts to yield independent event counts since the s offset time is greater than the ms gravitational-wave travel time between the sites plus the ms autocorrelation length of the templates. To test the independence of event counts over different time shifts over this observation period, we compute the differences in the number of background events having between consecutive time shifts. Figure 5 shows that the measured differences on these data follow the expected distribution for the difference between two independent Poisson random variables Skellam (1946), confirming the independence of time shifted event counts.
If a candidate event’s detection-statistic value is larger than that of any noise background event, as is the case for GW150914, then the PyCBC analysis places an upper bound on the candidate’s false alarm probability. After discarding time removed by data-quality vetoes and periods when the detector is in stable operation for less than seconds, the total observation time remaining is . Repeating the time-shift procedure times on these data produces a noise background analysis time equivalent to years. Thus, the smallest false alarm probability that can be estimated in this analysis is approximately . Since we treat the search parameter space as 3 independent classes, each of which may generate a false positive result, this value should be multiplied by a trials factor or look-elsewhere effect Lyons (2008) of 3, resulting in a minimum measurable false alarm probability of . The results of the PyCBC analysis are described in Sec. V.
Iv GstLAL Analysis
The GstLAL Gst (2016) analysis implements a time-domain matched filter search Cannon et al. (2012) using techinques that were developed to perform the near real-time compact-object binary searches Privitera et al. (2014); Messick et al. (2016). To accomplish this, the data and templates are each whitened in the frequency domain by dividing them by an estimate of the power spectral density of the detector noise. An estimate of the stationary noise amplitude spectrum is obtained with a combined median–geometric-mean modification of Welch’s method Messick et al. (2016). This procedure is applied piece-wise on overlapping Hann-windowed time-domain blocks that are subsequently summed together to yield a continuous whitened time series . The time-domain whitened template is then convolved with the whitened data to obtain the matched-filter SNR time series for each template. By the convolution theorem, obtained in this manner is the same as the obtained by frequency domain filtering in Eq. (1).
Of the 17.5 days of data that are used as input to the analysis, the GstLAL analysis discards times for which either of the LIGO detectors is in their observation state for less than s in duration. Shorter intervals are considered to be unstable detector operation by this analysis and are removed from the observation time. After discarding time removed by data-quality vetoes and periods when the detector operation is considered unstable the observation time remaining is days. To remove loud, short-duration noise transients, any excursions in the whitened data that are greater than are removed with 0.25 s padding. The intervals of vetoed in this way are replaced with zeros. The cleaned whitened data is the input to the matched filtering stage.
Adjacent waveforms in the template bank are highly correlated. The GstLAL analysis takes advantage of this to reduce the computational cost of the time-domain correlation. The templates are grouped by chirp mass and spin into 248 bins of templates each. Within each bin, a reduced set of orthonormal basis functions is obtained via a singular value decomposition of the whitened templates. We find that the ratio of the number of orthonormal basis functions to the number of input waveforms is 0.01 – 0.10, indicating a significant redundancy in each bin. The set of in each bin is convolved with the whitened data; linear combinations of the resulting time series are then used to reconstruct the matched-filter SNR time series for each template. This decomposition allows for computationally-efficient time-domain filtering and reproduces the frequency-domain matched filter to within 0.1% Cannon et al. (2010b, 2012, 2011).
Peaks in the matched-filter SNR for each detector and each template are identified over s windows. If the peak is above a matched-filter SNR of 4, it is recorded as a trigger. For each trigger, the matched-filter SNR time series around the trigger is checked for consistency with a signal by comparing the template’s autocorrelation function to the matched-filter SNR time series . The residual found after subtracting the autocorrelation function forms a goodness-of-fit test,
where is the time at the peak matched-filter SNR , and is a tunable parameter. A suitable value for was found to be 85.45 ms (175 samples at a