PRL116,061102(2016) PHYSICAL REVIEW LETTERS week ending 12 FEBRUARY 2016 multiple classes,this significance is decreased by a trials Detected with nc=20.0,GW150914 is the strongest factor equal to the number of classes [71]. event of the entire search.Consistent with its coalescence signal signature,it is found in the search class C3 of events A.Generic transient search with increasing time-frequency evolution.Measured on a Designed to operate without a specific waveform model, background equivalent to over 67 400 years of data and this search identifies coincident excess power in time- including a trials factor of 3 to account for the search frequency representations of the detector strain data classes,its false alarm rate is lower than I in 22 500 years. [43,72],for signal frequencies up to I kHz and durations This corresponds to a probability <2 x 10-6 of observing up to a few seconds. one or more noise events as strong as GW150914 during The search reconstructs signal waveforms consistent the analysis time,equivalent to 4.60.The left panel of with a common gravitational-wave signal in both detectors Fig.4 shows the C3 class results and background. using a multidetector maximum likelihood method.Each The selection criteria that define the search class C3 event is ranked according to the detection statistic reduce the background by introducing a constraint on the nc =V2Ec/(1+En/Ec),where Ec is the dimensionless signal morphology.In order to illustrate the significance of coherent signal energy obtained by cross-correlating the GW150914 against a background of events with arbitrary two reconstructed waveforms,and E,is the dimensionless shapes,we also show the results of a search that uses the residual noise energy after the reconstructed signal is same set of events as the one described above but without subtracted from the data.The statistic ne thus quantifies this constraint.Specifically,we use only two search classes: the SNR of the event and the consistency of the data the CI class and the union of C2 and C3 classes (C2+C3). between the two detectors. In this two-class search the GW150914 event is found in Based on their time-frequency morphology,the events the C2+C3 class.The left panel of Fig.4 shows the are divided into three mutually exclusive search classes,as C2+C3 class results and background.In the background described in [41]:events with time-frequency morphology of this class there are four events with ne>32.1,yielding a of known populations of noise transients(class C1),events false alarm rate for GW150914 of 1 in 8 400 years.This with frequency that increases with time(class C3),and all corresponds to a false alarm probability of 5x 10-6 remaining events (class C2) equivalent to 4.40. Generic transient search Binary coalescence search 2030 40 4.40 4.40 20304g 5.1 >5.10 2g304g4,60 >4.60 2a 30 405.10 >5.10 10 104 ■■■Search Result(C3) ■■■Search Result 101 Search Background (C3) 101 Search Background 100 ◆◆◆Search Result(C2+C3) 100 Background excluding GW150914 Search Background (C2+C3) 9 10-1 10- 10-2 GW150914 GW150914 10 10-3 10-4 10 10-5 10-6 10-6 10- 10- 10-8 10-8 12 14 16 18 20 532 10 18 20 24 Detection statistic nc Detection statistic pc FIG.4.Search results from the generic transient search (left)and the binary coalescence search(right).These histograms show the number of candidate events (orange markers)and the mean number of background events (black lines)in the search class where GW150914 was found as a function of the search detection statistic and with a bin width of 0.2.The scales on the top give the significance of an event in Gaussian standard deviations based on the corresponding noise background.The significance of GW150914 is greater than 5.1o and 4.60 for the binary coalescence and the generic transient searches,respectively.Left:Along with the primary search (C3)we also show the results (blue markers)and background (green curve)for an alternative search that treats events independently of their frequency evolution(C2+C3).The classes C2 and C3 are defined in the text.Right:The tail in the black-line background of the binary coalescence search is due to random coincidences of GW150914 in one detector with noise in the other detector.(This type of event is practically absent in the generic transient search background because they do not pass the time-frequency consistency requirements used in that search.)The purple curve is the background excluding those coincidences,which is used to assess the significance of the second strongest event. 061102-6
multiple classes, this significance is decreased by a trials factor equal to the number of classes [71]. A. Generic transient search Designed to operate without a specific waveform model, this search identifies coincident excess power in timefrequency representations of the detector strain data [43,72], for signal frequencies up to 1 kHz and durations up to a few seconds. The search reconstructs signal waveforms consistent with a common gravitational-wave signal in both detectors using a multidetector maximum likelihood method. Each event is ranked according to the detection statistic ηc ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Ec=ð1 þ En=EcÞ p , where Ec is the dimensionless coherent signal energy obtained by cross-correlating the two reconstructed waveforms, and En is the dimensionless residual noise energy after the reconstructed signal is subtracted from the data. The statistic ηc thus quantifies the SNR of the event and the consistency of the data between the two detectors. Based on their time-frequency morphology, the events are divided into three mutually exclusive search classes, as described in [41]: events with time-frequency morphology of known populations of noise transients (class C1), events with frequency that increases with time (class C3), and all remaining events (class C2). Detected with ηc ¼ 20.0, GW150914 is the strongest event of the entire search. Consistent with its coalescence signal signature, it is found in the search class C3 of events with increasing time-frequency evolution. Measured on a background equivalent to over 67 400 years of data and including a trials factor of 3 to account for the search classes, its false alarm rate is lower than 1 in 22 500 years. This corresponds to a probability < 2 × 10−6 of observing one or more noise events as strong as GW150914 during the analysis time, equivalent to 4.6σ. The left panel of Fig. 4 shows the C3 class results and background. The selection criteria that define the search class C3 reduce the background by introducing a constraint on the signal morphology. In order to illustrate the significance of GW150914 against a background of events with arbitrary shapes, we also show the results of a search that uses the same set of events as the one described above but without this constraint. Specifically, we use only two search classes: the C1 class and the union of C2 and C3 classes (C2 þ C3). In this two-class search the GW150914 event is found in the C2 þ C3 class. The left panel of Fig. 4 shows the C2 þ C3 class results and background. In the background of this class there are four events with ηc ≥ 32.1, yielding a false alarm rate for GW150914 of 1 in 8 400 years. This corresponds to a false alarm probability of 5 × 10−6 equivalent to 4.4σ. FIG. 4. Search results from the generic transient search (left) and the binary coalescence search (right). These histograms show the number of candidate events (orange markers) and the mean number of background events (black lines) in the search class where GW150914 was found as a function of the search detection statistic and with a bin width of 0.2. The scales on the top give the significance of an event in Gaussian standard deviations based on the corresponding noise background. The significance of GW150914 is greater than 5.1σ and 4.6σ for the binary coalescence and the generic transient searches, respectively. Left: Along with the primary search (C3) we also show the results (blue markers) and background (green curve) for an alternative search that treats events independently of their frequency evolution (C2 þ C3). The classes C2 and C3 are defined in the text. Right: The tail in the black-line background of the binary coalescence search is due to random coincidences of GW150914 in one detector with noise in the other detector. (This type of event is practically absent in the generic transient search background because they do not pass the time-frequency consistency requirements used in that search.) The purple curve is the background excluding those coincidences, which is used to assess the significance of the second strongest event. PRL 116, 061102 (2016) PHYSICAL REVIEW LETTERS week ending 12 FEBRUARY 2016 061102-6
week ending PRL116,061102(2016) PHYSICAL REVIEW LETTERS 12 FEBRUARY 2016 For robustness and validation,we also use other generic TABLE I.Source parameters for GW150914.We report transient search algorithms [41].A different search [73]and median values with 90%credible intervals that include statistical a parameter estimation follow-up [74]detected GW150914 errors,and systematic errors from averaging the results of with consistent significance and signal parameters. different waveform models.Masses are given in the source frame;to convert to the detector frame multiply by (1+z) [90].The source redshift assumes standard cosmology [91]. B.Binary coalescence search This search targets gravitational-wave emission from Primary black hole mass 36tM。 binary systems with individual masses from I to 99M Secondary black hole mass 294M。 total mass less than 100Mo,and dimensionless spins up to Final black hole mass 62M 0.99 [44].To model systems with total mass larger than Final black hole spin 4Mo,we use the effective-one-body formalism [75],which 0.671805 combines results from the post-Newtonian approach Luminosity distance 41010Mpc [11,76]with results from black hole perturbation theory Source redshift z 0.09888 and numerical relativity.The waveform model [77,78] assumes that the spins of the merging objects are aligned with the orbital angular momentum,but the resulting templates can,nonetheless,effectively recover systems When an event is confidently identified as a real with misaligned spins in the parameter region of gravitational-wave signal,as for GW150914,the back- GW150914 [44].Approximately 250 000 template wave- ground used to determine the significance of other events is reestimated without the contribution of this event.This is forms are used to cover this parameter space. The search calculates the matched-filter signal-to-noise the background distribution shown as a purple line in the ratio p(t)for each template in each detector and identifies right panel of Fig.4.Based on this,the second most maxima of p(t)with respect to the time of arrival of the signal significant event has a false alarm rate of I per 2.3 years and [79-81].For each maximum we calculate a chi-squared corresponding Poissonian false alarm probability of 0.02. statistic to test whether the data in several different Waveform analysis of this event indicates that if it is frequency bands are consistent with the matching template astrophysical in origin it is also a binary black hole [82].Values of near unity indicate that the signal is merger [44]. consistent with a coalescence.If is greater than unity.p(r) is reweighted asp=p/[1+()3]/2)1/6 [83.84].The final VI.SOURCE DISCUSSION step enforces coincidence between detectors by selecting The matched-filter search is optimized for detecting event pairs that occur within a 15-ms window and come from signals,but it provides only approximate estimates of the same template.The 15-ms window is determined by the the source parameters.To refine them we use general 10-ms intersite propagation time plus 5 ms for uncertainty in relativity-based models [77,78,87,88],some of which arrival time of weak signals.We rank coincident events based include spin precession,and for each model perform a on the quadrature sum pe of the p from both detectors [45]. coherent Bayesian analysis to derive posterior distributions To produce background data for this search the SNR of the source parameters [89].The initial and final masses, maxima of one detector are time shifted and a new set of final spin,distance,and redshift of the source are shown in coincident events is computed.Repeating this procedure Table I.The spin of the primary black hole is constrained ~107 times produces a noise background analysis time to be <0.7 (90%credible interval)indicating it is not equivalent to 608 000 years. maximally spinning,while the spin of the secondary is only To account for the search background noise varying across weakly constrained.These source parameters are discussed the target signal space,candidate and background events are in detail in [39].The parameter uncertainties include divided into three search classes based on template length. statistical errors and systematic errors from averaging the The right panel of Fig.4 shows the background for the results of different waveform models. search class of GW150914.The GW150914 detection- Using the fits to numerical simulations of binary black statistic value of Pe=23.6 is larger than any background hole mergers in [92,93],we provide estimates of the mass event,so only an upper bound can be placed on its false and spin of the final black hole,the total energy radiated alarm rate.Across the three search classes this bound is I in in gravitational waves,and the peak gravitational-wave 203000 years.This translates to a false alarm probability luminosity [39].The estimated total energy radiated in <2 x 10-7,corresponding to 5.10. gravitational waves is 3.0M.The system reached a A second,independent matched-filter analysis that uses a peak gravitational-wave luminosity of 3.6105 erg/s, different method for estimating the significance of its equivalent to /s. events [85,86],also detected GW150914 with identical Several analyses have been performed to determine signal parameters and consistent significance. whether or not GW150914 is consistent with a binary 061102-7
For robustness and validation, we also use other generic transient search algorithms [41]. A different search [73] and a parameter estimation follow-up [74] detected GW150914 with consistent significance and signal parameters. B. Binary coalescence search This search targets gravitational-wave emission from binary systems with individual masses from 1 to 99M⊙, total mass less than 100M⊙, and dimensionless spins up to 0.99 [44]. To model systems with total mass larger than 4M⊙, we use the effective-one-body formalism [75], which combines results from the post-Newtonian approach [11,76] with results from black hole perturbation theory and numerical relativity. The waveform model [77,78] assumes that the spins of the merging objects are aligned with the orbital angular momentum, but the resulting templates can, nonetheless, effectively recover systems with misaligned spins in the parameter region of GW150914 [44]. Approximately 250 000 template waveforms are used to cover this parameter space. The search calculates the matched-filter signal-to-noise ratio ρðtÞ for each template in each detector and identifies maxima of ρðtÞ with respect to the time of arrival of the signal [79–81]. For each maximum we calculate a chi-squared statistic χ2 r to test whether the data in several different frequency bands are consistent with the matching template [82]. Values of χ2 r near unity indicate that the signal is consistent with a coalescence. If χ2 r is greater than unity, ρðtÞ is reweighted as ρˆ ¼ ρ=f½1 þ ðχ2 r Þ3=2g1=6 [83,84]. The final step enforces coincidence between detectors by selecting event pairs that occur within a 15-ms window and come from the same template. The 15-ms window is determined by the 10-ms intersite propagation time plus 5 ms for uncertainty in arrival time of weak signals. We rank coincident events based on the quadrature sum ρˆ c of the ρˆ from both detectors [45]. To produce background data for this search the SNR maxima of one detector are time shifted and a new set of coincident events is computed. Repeating this procedure ∼107 times produces a noise background analysis time equivalent to 608 000 years. 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. The right panel of Fig. 4 shows the background for the search class of GW150914. The GW150914 detectionstatistic value of ρˆ c ¼ 23.6 is larger than any background event, so only an upper bound can be placed on its false alarm rate. Across the three search classes this bound is 1 in 203 000 years. This translates to a false alarm probability < 2 × 10−7, corresponding to 5.1σ. A second, independent matched-filter analysis that uses a different method for estimating the significance of its events [85,86], also detected GW150914 with identical signal parameters and consistent significance. When an event is confidently identified as a real gravitational-wave signal, as for GW150914, the background used to determine the significance of other events is reestimated without the contribution of this event. This is the background distribution shown as a purple line in the right panel of Fig. 4. Based on this, the second most significant event has a false alarm rate of 1 per 2.3 years and corresponding Poissonian false alarm probability of 0.02. Waveform analysis of this event indicates that if it is astrophysical in origin it is also a binary black hole merger [44]. VI. SOURCE DISCUSSION The matched-filter search is optimized for detecting signals, but it provides only approximate estimates of the source parameters. To refine them we use general relativity-based models [77,78,87,88], some of which include spin precession, and for each model perform a coherent Bayesian analysis to derive posterior distributions of the source parameters [89]. The initial and final masses, final spin, distance, and redshift of the source are shown in Table I. The spin of the primary black hole is constrained to be < 0.7 (90% credible interval) indicating it is not maximally spinning, while the spin of the secondary is only weakly constrained. These source parameters are discussed in detail in [39]. The parameter uncertainties include statistical errors and systematic errors from averaging the results of different waveform models. Using the fits to numerical simulations of binary black hole mergers in [92,93], we provide estimates of the mass and spin of the final black hole, the total energy radiated in gravitational waves, and the peak gravitational-wave luminosity [39]. The estimated total energy radiated in gravitational waves is 3.0þ0.5 −0.5M⊙c2. The system reached a peak gravitational-wave luminosity of 3.6þ0.5 −0.4 × 1056 erg=s, equivalent to 200þ30 −20M⊙c2=s. Several analyses have been performed to determine whether or not GW150914 is consistent with a binary TABLE I. Source parameters for GW150914. We report median values with 90% credible intervals that include statistical errors, and systematic errors from averaging the results of different waveform models. Masses are given in the source frame; to convert to the detector frame multiply by (1 þ z) [90]. The source redshift assumes standard cosmology [91]. Primary black hole mass 36þ5 −4M⊙ Secondary black hole mass 29þ4 −4M⊙ Final black hole mass 62þ4 −4M⊙ Final black hole spin 0.67þ0.05 −0.07 Luminosity distance 410þ160 −180 Mpc Source redshift z 0.09þ0.03 −0.04 PRL 116, 061102 (2016) PHYSICAL REVIEW LETTERS week ending 12 FEBRUARY 2016 061102-7