LIGO-G1900663-v3

TESTS OF GENERAL RELATIVITY WITH THE BINARY BLACK HOLE SIGNALS FROM THE LIGO-VIRGO CATALOGUE GWTC-1

N. K. Johnson-McDaniel (for the LIGO Scientific Collaboration and Virgo Collaboration) Recontres de Moriond - Gravitation 26.3.2019 Summary of arXiv:1903.04467

1 MOTIVATION 22 1822 GWTC-1 paper, LVC, arXiv:1811.12907 z 0.0 0.2 0.4 0.6 50 1.0 50 q 40 1/2 40 1/4

) 0.8 )

30 1/8 30 f a (M (M 2 20 m 20 0.6M 10 10

0 0.4 0 0 20 40 60 80 020 140 2 60 3 80 4 100 5 m1(M ) Mfd(ML (Gpc)) GW170817 GW151226 GW170104 GW170809 GW150914 GW170729 GW170817GW170608 GW151226GW151012 GW170104GW170814 GW170809GW170818 GW150914GW170823 GW170729 GW170608 GW151012 GW170814 GW170818 GW170823 10 binary black hole detections—variety of masses and distances FIG. 7. Parameter estimation summary plots IV. Posterior probability densities of distance d , inclination angle ✓ , and chirp mass of the Violent events (speeds up to ~0.5c and spacetime curvaturesL as small as ~14JN km) M FIG.GW 7. events. Parameter For estimation the two-dimensional summary plots distributions, IV. Posterior the contours probability show densities 90% credible of distance regions.d , inclination For GW170817 angle ✓ we, show and chirp results mass for theof high-spin the L JN M FIG. 4.GWprior events.Parameterai < For0.89. estimation theLeft two-dimensional panel: summaryThe inclination distributions, plots I. Posterior angle the and contours probability luminosity show densities distance 90% credible of of the the regions. binaries. masses, ForRight spins, GW170817 panel: and SNRThe we luminosity of show the results GW distance events. for the (or high-spin For redshift the z) two-dimensionaland source-frame distributions, chirp the mass. contours The colored show20 90% event credible labels are regions. orderedLeft by source-frame panel: Source chirp frame mass. component masses m1 and m2. We use the priorDistancesai < 0.89. Left from panel: EarthThe inclination ≳10 angle times and luminosity the binary’s distance ofown the binaries. lengthRight scale—test panel: The luminosity propagation distance (or redshifteffectsz) 2 convention that m m , which produces the sharp cut in the two-dimensional distribution. Lines of constant mass ratio q = m /m are shown and source-frame1 chirp2 mass. The colored event labels are ordered by source-frame chirp mass. 2 1 for 1/q = 2, 4, 8. For low-mass events, the contours follow lines of constant chirp mass. Right panel: The mass M f and dimensionless spin magnitude a f of the+ final1350 black holes. The colored event labels are ordered+0.19 by source frame chirp mass. The same color code and ordering dL = 2750 1320 Mpc, corresponding to a redshift of 0.48 0.20. mergers appear closer than they are and reduce their inferred, (where appropriate)+1350 apply to Figs. 5 to 8. +120 +0.19 redshift-corrected source frame masses, depending on the true dL The= 2750 closest1320 BBHMpc, is corresponding GW170608, toat d aL redshift= 320 of1100Mpc,.48 0.20 while. mergers appear closer than they are and reduce their inferred, ++12010 redshift-correcteddistance and magnification source frame factor masses, of the depending lens. Motivated on the true by the Thethe closest BNS BBH GW170817 is GW170608, was found at atdLd=L =3204011010 Mpc,Mpc. while The sig- nificant uncertainty in the luminosityd = distance+10 stems from the distanceheavy and BBHs magnification observed by factor LIGO of the and lens. Virgo, Motivated Ref. [187 by] theclaims wheretheM BNS= m GW170817+ m is the was total found mass at ofL the binary,40 10 Mpc. and m Theis sig- Binary neutron stars have additional degrees of freedom re- degeneracy1 2 between the distance and the binary’s inclination,1 heavythat BBHs four of observed the published by LIGO BBH and observations Virgo, Ref. [have187] been claims mag- definednificant to be uncertainty the mass of in the the larger luminosity component distance of stems the binary, from the lated to their response to a tidal field. The dominant quadrupo- degeneracyinferred from between the thesignal distance amplitude and the [128 binary’s, 184, 185 inclination,]. We show thatnified four of by the gravitational published lensing. BBH observations On the other have hand, been it hasmag- been such that m1 m2. Di↵erent parameterizations of spin e↵ects lar (` =pointed2) tidal out deformation that at LIGO’s is described and Virgo’s by the current dimensionless sensitivities it inferredjoint posteriors from the signal of luminosity amplitude distance [128, 184 and, 185 inclination]. We show✓JN in nified by gravitational lensing. On the other hand,5 it has been are possible and can be motivated from their appearance in is unlikely but⇤ not= impossible/ that2/ one/ of the GWs is multiply- jointthe posteriors left panel of in luminosityFig. 7. In general, distance the and inclination inclination angle✓JN isin onlytidalpointed deformability out that at LIGO’s(2 3) andk2 ( Virgo’sc G)(R currentm) of sensitivities each neu- it the GW phase or dynamics [121–123]. e↵ is approximately imaged. Ref. [188] suggests 10 5y 1 as a lower limit on the theweakly left panel constrained, in Fig. 7. In and general, for most the events inclination it has angle a bimodal is only dis-tronis star unlikely (NS), but where not impossiblek2 is the dimensionless that one of the` GWs= 2 Love is multiply- num- conserved throughout the inspiral [120]. To assess whether a ↵h 5 1 i weaklytribution constrained, around ✓JN and= for90 mostwith events greatest it hassupport a bimodal for the dis- sourceberimaged. andnumberR is Ref. the of NS [ BBH188 radius.] mergerssuggests The a 10 tidalected y deformabilities byas lensing, a lower whenlimit depend on consider- the binary is precessing we use a single e↵ective precession spin tributionbeing around either face✓ = on90 orwith face greatest o↵ (angular support momentum for the source pointedon thenumbering NS lensing mass of BBHm byand mergers clusters. the equation a↵ected of by state lensing, (EOS). when The consider- domi- parameter [124] (seeJN Appendix C). beingparallel eitherp or face antiparallel on or face to the o↵ line(angular of sight). momentum These orientations pointed nanting tidal lensingIn contribution the by right clusters. panel to the of Fig.GW7 phasewe show evolution the joint is encapsu- posterior be- During the inspiral the phase evolution depends at leading parallelproduce or antiparallel the greatest to gravitational-wave the line of sight). These amplitude orientations and so arelated inIntween an the e right↵ luminosityective panel tidal of distance deformability Fig. 7 (orwe redshift) show parameter the and joint source [ posterior130, frame131] be- chirp order on the chirp mass [34, 125, 126], produceconsistent the greatest with the gravitational-wave largest distance. amplitude For GW170817 and so the are✓JN tweenmass. luminosity We see thatdistance overall (or luminosity redshift) and distance source andframe chirp chirp mass 4 4 consistentdistribution with has the a largest single distance. mode.3/5 For For GW170809, GW170817 GW170818, the ✓JN are positively16 (m1 + correlated,12m2)m ⇤ as1 + expected(m2 + 12 form1 unlensed)m ⇤2 BBHs ob- (m1m2) mass.⇤˜ = We see that overall luminosity1 distance and2 chirp. mass(7) distributionand GW150914 has a single the= 90% mode. interval For, GW170809, contains only GW170818, a single(5) mode 5 M M1/5 areservations. positively13 correlated, as expectedM for unlensed BBHs ob- andso GW150914 that the 5th the percentile 90% interval lies above contains✓JN only= 90 a single. Orientations mode servations.An observed GW signal is registered with di↵erent arrival whichso is thatof also the the the total 5th best orbital percentile measured angular lies parameter above momentum✓ for= that low90 . are mass Orientations strongly sys- mis- JN times at the detector sites. The observed time↵ delays and am- temsof dominated thealigned total with by orbital the the inspiral angular line of [100 momentum sight, 121 are, 127 in that, general128 are]. strongly The disfavored mass mis- due An observed GW signal is registered with di erent arrival plitude and phase consistency of the signals at the sites allow ratio alignedto the with weaker the line emitted of sight GW are signal in general compared disfavored to observing due a times at the detector sites.B. The Masses observed time delays and am- plitudeus to and localize phase the consistency signal on of the the sky signals [189– at191 the]. sitesTwo detectorsallow to thebinary weaker face-on emitted (✓JN GW=m20 signal) or face-o compared↵ (✓JN to observing= 180). a For q = 1 (6) us tocan localize constrain the the signal sky on location the sky to [189 a broken–191]. annulus Two detectors [192–195] binaryGW170818 face-on the (✓JN misalignment= 0m) or face-o is more↵ (✓ likely,JN = 180 with). the For prob- In the left panel of Fig. 4 we show the inferred component 1 canand constrain the presence the sky of location additional to a broken detectors annulus in the [192 network–195] im- GW170818ability that the 45 misalignment < ✓JN < 135 is morebeing likely,0.38. with This the probability prob- masses of the binaries in the source frame as contours in the and e↵ective aligned spin e↵ appear in the phasing at higher andproves the presence localization of additional [19, 196– detectors198]. Fig. in8 theshows network the skyim- lo- abilityis less that than 45 0.36< ✓ forJN < all135 other being events.0.38 An. Thisinclination probability closem to -m plane. Because of the mass prior, we consider only sys- orders [100, 120, 122]. 1 proves2calizations localization for all [19 GW, 196 events.–198]. Both Fig. 8 panelsshows show the sky posteriors lo- is less✓JN than= 90 0.36would for enhance all other subdominant events. An inclination modes in the close GW to sig- m m tems within celestial1 coordinates2 and exclude which the shaded indicate region. the origin The of com- the sig- For✓ precessingnal,= 90 butwould also binaries result enhance the in orbital a subdominant weaker angular emitted modes momentum signal in the and, GW vec- to sig- com- calizations for all GW events. Both panels show posteriors JN ponentnal. masses In general, of the detected the credible BH regionsbinaries of cover sky positiona wide range are made tor ~Lnal,ispensate, not but a also stable a result closer direction, in source. a weaker and Ait emittedis more preferable precise signal to measurementand, describe to com- of in celestial coordinates which indicate the origin of the sig- fromnal.up In5M of general, ato collection70M the credible ofand disconnected lie regions within of the components sky range position expected determined are made for by the sourcepensate,the inclination inclination a closer by willsource. the be angle possible A more✓JN forbetween precise strongly measurement the precessing total an- of bina-stellar-mass⇠ BHs⇠ [132–134]. The posterior distribution of the ~ up ofthe a pattern collection of sensitivity of disconnected of the components individual detectors. determined The by top gularthe momentumries inclination [161, J186(which will]. be typically possible is for approximately strongly precessing constant bina- heavier component in the heaviest BBH, GW170729, grazes ries [161, 186]. ~ thepanel pattern shows of sensitivity localizations of the for individual confidently detectors. detected The O2 top events throughoutThis the inspiral) analysis andassumes the line that of the sight emitted vector GWN instead signal is notthe lowerthat boundary were communicated of the possible to EM mass observers gap expected and are from discussed ~ ~ panel shows localizations for confidently detected O2 events of the orbitalThisa↵ected analysis inclination by gravitational assumes angle that◆ between lensing. the emittedL Lensingand GWN [ would118 signal, 129 make is]. not GWpulsationalthatfurther were pair communicated in Ref. instability [22]. The and to EM results pair observers instability for the and credible supernovae are discussed regions at and We quote↵ frequency-dependent quantities such as spin vec- 60 120M [135–137]. The lowest-mass BBH systems, a ected by gravitational lensing. Lensing would make GW ⇠ further in Ref. [22]. The results for the credible regions and tors and derived quantities as p at a GW reference frequency GW151226 and GW170608, have 90% credible lower bounds fref = 20Hz. on m2 of 5.6 M and 5.9 M , respectively, and therefore lie OVERVIEW

No detailed predictions for waveforms from binary black hole coalescences in alternative theories of gravity ⇒ Perform null tests

Four different tests:

Residuals

Inspiral-merger-ringdown consistency

Parameterized tests of GW generation

Parameterized tests of GW propagation

Also check constraints on GW polarizations

We find no evidence for deviations from GR’s predictions. 3 EVENT SELECTION

Only consider BBHs—the BNS GW170817 has its own testing GR paper from the LVC (arXiv:1811.00364)

Individual results presented for FARs < 1/yr in any pipeline (modeled or weakly modeled)

Use confident events to obtain combined results:

False alarm rate (FAR) < 1/(1000 yr) in both modeled pipelines

All results publicly available for further exploration: https://dcc.ligo.org/LIGO-P1900087/public 4 4 4

TABLE I. The GW events considered in thisTABLE paper, I. separated The GW by events observing considered run. The in this first paper, block separated of columns by gives observing the names run. The of the first events block and of columns gives the names of the events and lists some of their relevant properties obtainedlists using some GR of their waveforms relevant (luminosity properties obtained distance usingDL, source GR waveforms frame total (luminosity mass Mtot and distance final massDL, sourceMf, frame total mass Mtot and final mass Mf, and dimensionless final spin af). The next blockand dimensionless of columns gives final the spin significance,af). The next measured block of by columns the false-alarm-rate gives the significance, (FAR), with measured which byeach the false-alarm-rate (FAR), with which each event was detected by each of the three searchesevent was employed, detected as by well each as of the the matched three searches filter signal-to-noise employed, as ratio well from as the the matched stochastic filter sampling signal-to-noise ratio from the stochastic sampling analyses with GR waveforms. A dash indicatesanalyses that withan event GR was waveforms. not identified A dash by indicates a search. that The an parameters event was and not identifiedSNR values by give a search. the medians The parameters and SNR values give the medians and 90% credible intervals. All the events exceptand 90% for credibleGW151226 intervals. and GW170729 All the events are consistent except for with GW151226 a binary and of nonspinning GW170729 are black consistent holes (when with a binary of nonspinning black holes (when analyzed assuming GR). See [14] for moreanalyzed details about assuming all the GR). events. Seea The [14] last for block more detailsof columns about indicates all the events. whicha GRThe tests last areblock performed of columns on a indicates which GR tests are performed on a given event: RT = residuals test (Sec. VA);given IMREVENT= event:inspiral-merger-ringdown RT = residuals SELECTION test (Sec. consistencyVA); IMR test= (Sec.inspiral-merger-ringdownVB); PI & PPI = parameterized consistency tests test of (Sec. GWVB); PI & PPI = parameterized tests of GW generation for inspiral and post-inspiral phasesgeneration (Sec. VI for); MDRinspiral= andmodified post-inspiral GW dispersion phases (Sec. relationVI); (Sec. MDRVII=).modified The events GW with dispersion bold names relation are (Sec. VII). The events with bold names are used to obtain the combined results for eachused test. to obtain the combined results for each test. PropertiesProperties FAR GR FAR tests performed GR tests performed Event Event SNR SNR DL Mtot Mf af PyDCBCL Mtot GstLALMf cWBaf PyCBC RT IMR GstLAL PI PPI MDRcWB RT IMR PI PPI MDR 1 1 1 1 Events1 in boldface1 used [Mpc] [M ][M ][Mpc] [yr ] [M ][ [yr ]M ] [yr ] [yr ] [yr ] [yr ] for combined results +150 +3.7 +3.3 +0.05 +150 5 +3.7 7+3.3 +0.05 4 +0.1 533337 3 4 +0.1 3333 3 GW150914 430 170 66.2 3.3 63.1 3.0GW1509140.69 0.04 <4301.5 1701066<.213.0.3 6310.1 3<.0 10.6.69 100.04 25<.13.50.2 10 < 1.0 10 < 1.6 10 25.3 0.2 ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ +550 +10.6 +10.7 +0.13 +550 +10.6 3+10.7 +0.13 +0.3 3 3 33 +0.3 3 33 GW151012 1060 480 37.3 3.9 35.7 3.8GW1510120.67 0.11 10600.17480 37.37.39.9 1035.7 3.8 0.67– 0.11 9.2 0..417 7.9––10 – 9.2 0.4 –– ⇥ ⇥ +180 +6.2 +6.4 +0.07 +180 5 +6.2 7+6.4 +0.07 +0.2 53 3 7 3 +0.2 3 3 3 GW151226 440 190 21.5 1.5 20.5 1.5GW1512260.74 0.05 <4401.7 1901021<.511.0.5 2010.5 1.5 00.74.020.05 12<.14.70.3 10 < 1–.0 10 – 0.02 12.4 0.3 – – ⇥ ⇥ ⇥ ⇥ +440 +5.3 +5.2 +0.08 +440 5 +5.3 7+5.2 +0.084 +0.2 533337 3 4 +0.2 3333 3 GW170104 960 420 51.3 4.2 49.1 4.0GW1701040.66 0.11 <9601.4 4201051<.314.0.2 4910.1 4.02.09.66100.11 14<.10.40.3 10 < 1.0 10 2.9 10 14.0 0.3 ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ +120 +3.1 +3.2 +0.04 +120 4 +3.1 7+3.2 +0.044 +0.2 43 337 3 4 +0.2 3 33 3 GW170608 320 110 18.6 0.7 17.8 0.7GW1706080.69 0.04 <3203.1 1101018<.610.0.7 1710.8 0.71.04.69100.04 15<.36.10.3 10 < 1–.0 10 1.4 10 15.6 0.3 – ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ +1380 +15.6 +14.6 +0.07 +1380 +15.6 +14.6 +0.07 +0.4 33 Highest33 significance+0.4 33 33 GW170729 2760 1340 85.2 11.1 80.3 10.GW1707292 0.81 0.13 27601.41340 85.2 110.1.1880.3 10.2 00.81.020.13 10.8 10..54 0.18– 0.02 10.8 0.5 – +320 +5.4 +5.2 +0.08 +320 4 +5.4 7+5.2 +0.08 +0.2 4 33 in 7unmodeled33 search+0.2 33 33 GW170809 990 380 59.2 3.9 56.4 3.7GW1708090.70 0.09 1990.4 38010 59<.213.0.9 5610.4 3.7 0.70– 0.09 121..74 0.310 < 1.0 10– – 12.7 0.3 – ⇥ ⇥ ⇥ ⇥ +160 +3.4 +3.2 +0.07 +160 5 +3.4 7+3.2 +0.07 4 +0.3 533337 3 4 +0.3 3333 3 GW170814 580 210 56.1 2.7 53.4 2.4GW1708140.72 0.05 <5801.2 2101056<.112.0.7 5310.4 2<.4 20.1.72 100.05 17<.18.20.3 10 < 1.0 10 < 2.1 10 17.8 0.3 ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ +430 +5.1 +4.8 +0.07 +430 +5.1 5 +4.8 +0.07 +0.3 33 5 33 +0.3 33 33 GW170818 1020 360 62.5 4.0 59.8 3.8GW1708180.67 0.08 1020–360 62.54.24.0 1059.8 3.8 0.67– 0.08 11.9 0.–4 4.2 10– – 11.9 0.4 – ⇥ ⇥ +840 +9.9 +9.4 +0.08 +840 5 +9.9 7+9.4 +0.083 +0.2 533 7333 +0.2 33 33 GW170823 1850 840 68.9 7.1 65.6 6.6GW1708230.71 0.10 <18503.3 8401068<.917.0.1 6510.6 6.62.01.71100.10 12<.31.30.3 10 < 1.0 10– 2.1 10 12.1 0.3 – ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ a The parameters given in this table di↵er slightlya fromThe parameters those in v2 given of the in arXiv this table version di↵ ofer [ slightly14] since from they those correct in v2 for of a smallthe arXiv issue version in the applicationof [14] since of they priors correct for for a small issue in the application of priors for spin parameters. spinUse parameters. Looks for chirping signals modeled weakly without using any GR predictions GR templates modeled 5 285 respect to the detector network. The285 intrinsicrespect to parameters the detector for network.309 aligned-spin The intrinsic model. parameters We use IMRP for henom309 aligned-spinPv2 to obtain model. all the We use IMRPhenomPv2 to obtain all the 286 circularized black-hole binaries in GR286 arecircularized the two masses black-holemi of binaries310 main in results GR are given the two in this masses paper,mi of and310 usemainSEOBNR resultsv4 givento check in this paper, and use SEOBNRv4 to check ~ 287 the black holes and the two spin vectors287 theS~ i defining black holes the and rotation the two311 spinthe robustness vectors S i defining of these the results, rotation whenever311 the possible. robustness When of these we results, whenever possible. When we 288 of each black hole, where i 1, 2 labels288 of each the two black black hole, holes. where312i use1IMRP, 2 labelshenom thePv two2, we black impose holes. a312 priorusemIMRP/m henom18 onPv2 the, we impose a prior m /m 18 on the 2 { } 2 { } 1 2  1 2  289 We assume that the binary has negligible289 We orbital assume eccentricity, that the binary313 hasmass negligible ratio, as theorbital waveform eccentricity, family313 ismass not calibrated ratio, as the against waveform family is not calibrated against 290 as is expected to be the case when the290 binaryas is expected enters the to band be the of case314 numerical when the binary relativity enters simulations the band of for314m1numerical/m2 > 18. relativity We do not simulations for m1/m2 > 18. We do not 291 ground-based detectors [45, 46] (except291 ground-based in some more detectors extreme [45315 ,impose46] (except a similar in some prior more when extreme using315SEOBNRimposev a4 similar, since itprior in- when using SEOBNRv4, since it in- 4 4 292 formation scenarios, e.g., [53–56]).292 Theformation extrinsic scenarios, parameterse.g.,316 [cludes53–56]). information The extrinsic about parameters the extreme316 masscludes ratio information limit. Neither about the extreme mass ratio limit. Neither 293 comprise four parameters that specify293 thecomprise space-time four parameters location 317 thatof these specify waveform the space-time models location includes317 theof these full spin waveform dynamics models includes the full spin dynamics 294 of the binary black-hole, namely the294 skyof location the binary (right black-hole, ascen- 318 namely(which the requires sky location 6 spin (rightparameters). ascen- Fully318 (which precessing requires waveform 6 spin parameters). Fully precessing waveform 295 sion and declination), the luminosity295 distance,sion and and declination), the time of the319 luminositymodels have distance, been recently and the time developed of 319 [models21, 59–61 have] and been will recently be developed [21, 59–61] and will be 296 coalescence. In addition, there are three296 coalescence. extrinsic parameters In addition,320 thereused are in future three extrinsic applications parameters of these tests.320 used in future applications of these tests.

297 that determine the orientation of the297 binarythat determine with respect the orientation to 321 The of waveform the binary models with respect used in to this321 paperThe do waveform not include models the used in this paper do not include the 298 Earth, namely the inclination angle298 ofEarth, the orbit namely with the respect inclination322 e↵ects angle of of subdominant the orbit with (non-quadrupole) respect 322 e↵ects modes, of subdominant which are (non-quadrupole) modes, which are 299 to the observer, the polarization angle,299 andto the the observer, orbital phase the polarization at 323 expected angle, to and be the small orbital for comparable-mass phase at 323 expected binaries to be [62 small, 63]. for comparable-mass binaries [62, 63]. 300 coalescence. 300 coalescence. 324 The first generation of binary black324 holeThe waveform first generation models of binary black hole waveform models 301 We employ two waveform families301 thatWe model employ binary two black waveform325 including families spin that model and higher binary order black modes325 including has recently spin been and higher de- order modes has recently been de- 302 302 v v holes in GR: the e↵ective-one-body basedholesSEOBNR in GR: the4 e[↵18ective-one-body] 326 veloped [61 based, 64–SEOBNR66]. Preliminary4 [18] results326 veloped in [14 [61],, using64–66 NR]. Preliminary results in [14], using NR 303 waveform family that assumes non-precessing303 waveform spins family for that the assumes327 simulations non-precessing supplemented spins byfor NR the surrogate327 simulations waveforms, supplemented indi- by NR surrogate waveforms, indi- 304 black holes (we use the frequency304 domainblack holes reduced (we order use the328 frequencycate that the domain higher reduced mode content order of328 thecate GW that signals the higher detected mode content of the GW signals detected 305 v 305 v model SEOBNR 4 ROM for reasonsmodel of computationalSEOBNR 4 eROM- 329 forby Advanced reasons of LIGO computational and Virgo e is- weak329 by enough Advanced that LIGO models and Virgo is weak enough that models 306 306 he he ciency), and the phenomenological waveformciency), family and theIMRP phenomenological- 330 without waveform the e↵ect of family subdominantIMRP - modes330 without do not the introduce e↵ect of sub- subdominant modes do not introduce sub- 307 nom v 307 nom v P 2 [19, 57, 58] that models the e↵ectsP of2 precessing[19, 57, 58 spins] that331 modelsstantial the biases e↵ects in of the precessing intrinsic spins parameters331 stantial of the biases binary. in the For intrinsic parameters of the binary. For 308 using two e↵ective parameters by twisting308 using up two the e↵ underlyingective parameters332 unequal-mass by twisting binaries, up the theunderlying e↵ect of the332 unequal-mass non-quadrupole binaries, modes the e↵ect of the non-quadrupole modes 333 is more pronounced [67], particularly333 is when more the pronounced binary’s ori- [67], particularly when the binary’s ori- 334 entation is close to edge-on. In these334 entation cases, the is presence close to edge-on. of In these cases, the presence of 4 These scenarios could occur often enough, compared4 These to scenarios the expected could rate occur335 oftennon-quadrupole enough, compared modes to the expected can show rate up335 asnon-quadrupole a deviation from modes GR can show up as a deviation from GR of detections, that the inclusion of eccentricityof in detections, waveform models that the is inclusion a 336 when of eccentricity using waveforms in waveform that models only is include a 336 when the quadrupole using waveforms modes, that only include the quadrupole modes, necessity for tests of GR in future observing runs;necessity see, e.g., for tests [47–52 of] GR for in337 futureas was observing shown runs; in [ see,68]. e.g., Applications [47–52] for of337 testsas was of GR shown with in the [68 new]. Applications of tests of GR with the new recent work on developing such waveform models.recent work on developing such waveform models. 338 waveform models that include non-quadrupole338 waveform modes models will that be include non-quadrupole modes will be Class. Quantum Grav. 35 (2018) 014002 A Ghosh et al

Class. Quantum Grav. 35 (2018) 014002 A Ghosh et al

COMBINING RESULTS

GR injections Combining together results lets us improve accuracy, roughly as N-1/2.

Naïvely combining together results assuming the parameters are the same in all cases only gives conservative constraints assuming GR is true (or if dependence on binary’s parameters is taken into account in test, like distance in propagation test) non-GR injections Nevertheless, we combine naïvely, in keeping with the spirit of our null tests, and because individual results show no evidence for departures from GR.

Additionally, it is possible to pick up some departures from GR with naïve combination.

[Illustration for the IMR consistency test from Ghosh et al. CQG (2018)] 6 Figure 3. Left: In each plot shaded regions show the 68% and 95% credible intervals on theFigure combined 8. Same posteriors as fgure on ϵ3 := except∆Mf / thatM¯ f , theσ : = test∆ a isf / a¯ performedf from multiple on simulated observations signals of GRcontaining signals, aplotted modif againstcation fromthe number GR described of binary in black section hole 4.2 observations.. The combined The GRposteriors value (fromϵ = σ multiple= 0) is observationsindicated by showhorizontal a clear dashed departure lines. from The the mean GR predictionsvalue of the (horizontal posterior fromdashed each lines event on theis shown left plots as an and orange the plus dot sign along on with the rightthe corresponding plots). 68% credible interval as an orange vertical line. Posteriors on ε are marginalized over σ, and vice the waveformsversa. are generated Right: the withthin orange a modi contoursfcation show of the the GR 68% energy credible f uxregions as described of the individual in sec- tion 4.1. Also, posteriorsfor simplicity, on the we ϵ, σconsider computed binary from theblack same holes events. with The zero GR spins, value since(ϵ = σthe= 0EOB) is waveform familyindicated that we by employ the black to +produce sign. Right modi inset:fed GRThe waveformsred contours is show a nonspinning the 68% credible model [42, 43]. However,regions we on still the perform combined parameter posterior fromestimation 5, 10 and using 25 events the same (with SEOBNRv2_ROM_ increasing shades of Double Spin aligned-spindarkness). The model GR value employed (ϵ = σ in= 0section) is indicated 3. by the black + sign. Different rows correspond to different catalogs of 50 randomly chosen events from a total of ∼100 Modifed GRsimulated waveforms events. for binaries with different mass ratios (as well as distances, sky locations, and other extrinsic parameters) were constructed using the prescription presented 10

15 WAVEFORMS & INFERENCE

All tests rely on waveform models accurately describing predictions of GR

Use precessing IMRPhenomPv2 waveforms for primary results and aligned-spin SEOBNRv4_ROM waveforms as a check on waveform systematics.

Don’t include: Two-spin effects, higher modes, eccentricity.

Expect such effects to be negligible for these signals, but future applications of these tests will include them—waveform models are now available/being developed

[e.g., Khan et al. arXiv (2018) (two-spin phenom); London et al. PRL (2018) (higher-mode phenom), Cotesta et al. PRD (2018) (higher-mode EOB), Varma et al. arXiv (2018) (higher-mode NR hybrid surrogate); Huerta et al. PRD (2018) (eccentric IMR), Moore and Yunes arXiv (2019) (eccentric PN)]

Use standard Bayesian inference methods [LALInference, Veitch et al. PRD (2015)] to infer binary parameters (and non-GR parameters, when appropriate) from data 7 week ending PRL 118, 221101 (2017) PHYSICAL REVIEW LETTERS 2 JUNE 2017

calibration uncertainties for strain data in both detectors for the times used in this analysis are better than 5% in amplitude and 3° in phase over the frequency range 20– 1024 Hz. At the time of GW170104, both LIGO detectors were operating with sensitivity typical of the observing run to date and were in an observation-ready state. Investigations similar to the detection validation procedures for previous events [2,14] found no evidence that instrumental or environmental disturbances contributed to GW170104. RESIDUALS TEST: FORMULATION III. SEARCHES GW170104 was first identified by inspection of low- latency triggers from Livingston data [15–17]. An auto- mated notification was not generated as the Hanford detector’s calibration state was temporarily set incorrectly in the low-latency system. After it was manually deter- mined that the calibration of both detectors was in a nominal state, an alert with an initial source localization Check if there is any power in the [18,19] was distributed to collaborating astronomers [20] for the purpose of searching for a transient counterpart. data not captured by a GR waveform. About 30 groups of observers covered the parts of the sky localization using ground- and space-based instruments, spanning from γ ray to radio frequencies as well as high- energy neutrinos [21]. Offline analyses are used to determine the significance of Subtract maxL waveform from data week ending candidate events. They benefit from improved calibration PRL 118, 221101 (2017) PHYSICAL REVIEW LETTERS 2 JUNE 2017 and refined data quality information that is unavailable to and compute SNR of residuals using low-latency analyses [5,14]. The second observing run is Effectively all of the information comes from constraints on FIG. 1. Time–frequency representation [9] of strain data from divided into periods of two-detector cumulative coincident χ combined with the mass ratio (and our prior ofHanford and Livingston detectors (top two panels) at the time of eff observing time with 5 days of data to measure the false unmodeled waveletisotropically analysis distributed orientations and uniformly distrib-GW170104. The data begin at 1167559936.5 GPS time. The ≳ uted magnitudes) [5]. third panel from the top shows the time-series data from each alarm rate of the search at the level where detections can be confidently claimed. Two independently designed matched (BayesWave) The source’s luminosity distance D L is inferred from thedetector with a 30–350 Hz bandpass filter, and band-reject filters signal amplitude [37,74]. The amplitude is inversely propor-to suppress strong instrumental spectral lines. The Livingston filter analyses [16,22] used 5.5 days of coincident data tional to the distance, but also depends upon the binary’datas have been shifted back by 3 ms to account for the source’s collected from January 4, 2017 to January 22, 2017. inclination [59,75–77]. This degeneracy is a significantsky location, and the sign of its amplitude has been inverted to These analyses search for binary coalescences over a range source of uncertainty [57,71]. The inclination has a bimodalaccount for the detectors’ different orientations. The maximum- of possible masses and by using discrete banks [23–28] of Compare to SNRsdistribution of background with broad peaks for face-on and face-offlikelihood binary black hole waveform given by the full-pre- waveform templates modeling binaries with component orientations (see Fig. 4 of the Supplemental Materialcession model (see Sec. IV) is shown in black. The bottom panel spins aligned or antialigned with the orbital angular momen- [11]). GW170104’s source is at D 880 450 Mpc, corre-shows the residuals between each data stream and the maximum- data around each event L −þ390 tum [29]. The searches can target binary black hole mergers ¼ 0.08 likelihood waveform. det sponding to a cosmological redshift of z 0.18þ0.07 [52]. with detector-frame total masses 2M M 100–500M , ¼ − ≤ While GW170104’s source has masses and spins and spin magnitudes up to ∼0.99. The⊙ upper≲ mass boundary⊙ GW170104 residuals and BayesWavecomparable to reconstruction GW150914’s, it is most probably atAfter a the first observing run, both LIGO detectors under- of the bank is determined by imposing a lower limit on the greater distance [5,37]. of signal from detection paper, LVC, PRL (2017) went commissioning to reduce instrumental noise, and 8 to duration of the template in the detectors’ sensitive frequency For GW150914, extensive studies were made to verify theimprove duty factor and data quality (see Sec. I in the band [30]. Candidate events must be found in both detectors accuracy of the model waveforms for parameter estimation SupplementalFIG. 4. Time-domain Material [11] detector). For data the (gray), Hanford and 90% detector, confidence a through comparisons with numerical-relativity waveforms by the same template within 15 ms [4]. This 15-ms window is high-powerintervals laser for waveformsstage was reconstructedintroduced, and from as the the morphology- first step determined by the 10-ms intersite propagation time plus an [78,79]. GW170104 is a similar system to GW150914 and, independent wavelet analysis (orange) and binary black hole therefore, it is unlikely that there are any significant biases inthe laser(BBH) power models was from increased both waveform from families 22 to 30 (blue), W whitenedto reduce by allowance for the uncertainty in identified signal arrival times our results as a consequence of waveform modeling. Theshot noiseeach instrument[10] at’s high noise frequencies. amplitude spectral For the density. Livingston The left of weak signals. Candidate events are assigned a detection lower SNR of GW170104 makes additional effects notdetector,ordinate the laseraxes are power normalized was unchanged, such that the but amplitude there was of the a statistic value ranking their relative likelihood of being a incorporated in the waveform models, such as higher modessignificantwhitened improvement data and the physical in low-frequency strain are equal performance at 200 Hz. The gravitational-wave signal: the search uses an improved [55,80,81], less important. However, if the source is edge onmainlyright due ordinate to the axes mitigation are in units of of scattered noise standard light deviations. noise. The detection statistic compared to the first observing run [31]. or strongly precessing, there could be significant biases in width of the BBH region is dominated by the uncertainty in the Calibrationastrophysical of parameters. the interferometers is performed by The significance of a candidate event is calculated by quantities including M and χeff [78]. Comparison toinducing test-mass motion using photon pressure from comparing its detection statistic value to an estimate of numerical-relativity simulations of binary black holes with modulatedbetween calibration the maximum-likelihood lasers [12,13] waveform. The of one-sigma the binary the background noise [4,16,17,22]. GW170104 was detected nonprecessing spins [79], including those designed to black hole model and the median waveform of the morphol- replicate GW170104, produced results (and residuals) con- ogy-independent analysis is 87%, consistent with expect- sistent with the model-waveform analysis. ations from Monte Carlo analysis of binary black hole signals221101-2 injected into detector data [34]. We also use the morphology- V. WAVEFORM RECONSTRUCTIONS independent analysis to search for residual gravitational- Consistency of GW170104 with binary black hole wave- wave energy after subtracting the maximum-likelihood form models can also be explored through comparisons with binary black hole signal from the measured strain data. a morphology-independent signal model [82]. We choose to There is an 83% posterior probability in favor of Gaussian describe the signal as a superposition of an arbitrary number noise versus residual coherent gravitational-wave energy of Morlet-Gabor wavelets, which models an elliptically which is not described by the waveform model, implying that polarized, coherent signal in the detector network. GW170104’s source is a black hole binary. Figure 4 plots whitened detector data at the time of GW170104, together with waveforms drawn from the VI. BINARY BLACK HOLE POPULATIONS 90% credible region of the posterior distributions of the AND MERGER RATES morphology-independent model and the binary black hole waveform models used to infer the source properties. The The addition of the first 11 days of coincident observing signal appears in the two detectors with slightly different time in the second observing run, and the detection of amplitudes, and a relative phase shift of approximately 180°, GW170104, leads to an improved estimate of the rate because of their different spatial orientations [2]. The wave- density of binary black hole mergers. We adopt two simple let- and template-based reconstructions differ at early times representative astrophysical population models: a distribu- because the wavelet basis requires high-amplitude, well- tion that is a power law in m 1 and uniform in m 2, −α localized signal energy to justify the presence of additional p m 1;m2 ∝ m 1 = m 1 − 5M with α 2.35 [83], and ð Þ ð ⊙Þ ¼ wavelets, while the earlier portion of the signal is inherently a distribution uniform in the logarithm of each of the included in the binary black hole waveform model. component masses [5,8]. In both cases, we impose The waveforms reconstructed from the morphology- m 1;m2 ≥ 5M and M ≤ 100M [8]. Using the results independent model are consistent with the characteristic from the first⊙ observing run as a⊙ prior, we obtain updated 110 3 1 inspiral-merger-ringdown structure. The overlap [58] rates estimates of R 103þ63 Gpc− yr− for the power ¼ −

221101-5 6

446 distribution function. Fig. 1 also displays the actual value of TABLE II. Results of the residuals analysis. For each event, this table 447 SNR measured from the residuals of each event (dotted ver- presents the 90%-credible upper limit on the reconstructed network 90 448 tical line). In each case, the height of the curve evaluated at SNR after subtraction of the best-fit GR waveform (SNR90), a cor- 449 the SNR measured for the corresponding detection yields the responding lower limit on the fitting factor (FF90 in the text), and 90 450 the SNR90 p-value. SNR90 is a measure of the maximum possible p-value reported in Table II (markers in Fig. 1). coherent signal power not captured by the best-fit GR template, while 451 The values of residual SNR90 vary widely among events the p-value is an estimate of the probability that instrumental noise 452 because they depend on the specific state of the instruments produced such SNR90 or higher. We also indicate which interferome- 453 at the time of detection: segments of data with elevated noise ters (IFOs) were used in the analysis of a given event, either the two 454 levels are more likely to result in spurious coherent residual Advanced LIGO detectors (HL) or the two Advanced LIGO detectors 455 power, even if the signal agrees with GR. In particular, the plus Advanced Virgo (HLV). See Sec. VAin the main text for details. 456 background distributions for events seen by three detectors are

Event IFOs Residual SNR90 Fitting factor p-value 457 qualitatively di↵erent from those seen by only two. This is both 458 ayes ave GW150914 HL 6.4 0.97 0.34 due to (i) the fact that B W is configured to expect the 459 GW151012 HL 6.9 0.81 0.18 SNR to increase with the number of detectors and (ii) the fact GW151226 HL 5.7 0.91 0.76 460 that Virgo data present a higher rate of non-Gaussianities than 461 LIGO. We have confirmed both these factors play a role by GW170104 HL 5.2 0.94 0.97 462 studying the background SNR90 distributions for real data from GW170608 HL 7.8 0.90 0.07 463 GW170729 HLV 6.5 0.87 0.72 each possible pair of detectors, as well as distributions over 464 GW170809 HLV 6.7 0.91 0.73 fabricated Gaussian noise. Specifically, removing Virgo from GW170814 HLV 8.6 0.90 0.19 465 the analysis results in a reduction in the coherent residual power HL HL RESIDUALS TEST: RESULTS466 GW170818 HLV 10.1 0.78 0.13 for GW170729 (SNR90 = 6.0), GW170809 (SNR90 = 6.3), HL HL GW170823 HL 5.4 0.92 0.89 467 GW170814 (SNR = 5.9), and GW170818 (SNR = 6.6). 90 90 468 The event-by-event variation of SNR90 is also reflected in 469 the values of FF90. GW150914 provides the strongest result 470 minimumwith FF90 = 0p-value:.97, which 0.07 corresponds to an upper limit of 3% 471 on the magnitude of potential deviations from our GR-based (GW170608)7 472 template, in the specific sense defined in [4] and discussed 473 above. On the other hand, GW170818 yields the weakest result 474 with FF = 0.78 and a corresponding upper limit on waveform meta p-value90 [using Fisher’s 475 mismatch of 22%. The average FF90 over all events is 0.89. 476 method]The set: of p-values shown in Table II is consistent with all 477 0.4;coherent no evidence residual power against being due to instrumental noise. As- 478 uniformlysuming that thisdistributed is indeed the p- case, we expect the p-values to 479 be uniformly distributed over [0, 1], which explains the vari- 480 valuesation in Table II. With only ten events, however, it is dicult 481 to obtain strong quantitative evidence of the uniformity of this 482 Thus,distribution. no evidence Nevertheless, for we follow Fisher’s method [74] to 483 compute a meta p-value for the null hypothesis that the individ- 484 coherentual p-values power in Table II notare uniformly distributed. We obtain a 485 attributablemeta p-value of to 0.4, noise implying that there is no evidence for co- 486 herent residual power that cannot be explained by noise alone. GW170608 9 487 All in all, this means that there is no statistically significant FIG. 1. Survival function (p = 1 CDF) of the 90%-credible upper 488 evidence for deviations from GR. limit on the network SNR (SNR90) for each event (solid or dashed curves), compared to the measured residual values (vertical dotted lines). For each event, the value of the survival function at the mea- 489 B. Inspiral-merger-ringdown consistency test sured SNR90 gives the p-value reported in Table II (markers). The colored bands correspond to uncertainty regions for a Poisson pro- cess and have half width p/ pN, with N the number of noise-only 490 The inspiral-merger-ringdown consistency test for binary ± n instantiations that yielded SNR90 greater than the abscissa value. 491 black holes [38, 75] checks the consistency of the low- 492 frequency part of the observed signal (roughly correspond- 493 ing to the inspiral of the black holes) with the high-frequency

439 Our results are summarized in Table II. For each event, 440 we present the values of the residual SNR90, the lower limit 441 on the fitting factor (FF ), and the SNR p-value. The 90 90 7 This value is better than the one quoted in [4] by 1 percentage point. The 442 background distributions that resulted in those p-values are small di↵erence is explained by several factors, including that paper’s use 443 shown in Fig. 1. In Fig. 1 we represent these distributions of the maximum a posteriori waveform (instead of maximum likelihood) 444 through the empirical estimate of their survival functions, i.e., and 95% (instead of 90%) credible intervals, as well as improvements in data calibration. 445 p(SNR ) = 1 CDF(SNR ), with “CDF” the cumulative 90 90 2

l3 T = eff , S 36 c where

⌦ 1 2 m 0.14. eff ⌘ ⌦ ⇡ ✓ ◆

I. TYPESETTING FOR TALKS

MKepler 2⇡⇢ 2 ⌦m 2 2 + Cmatch = rmatch = H rmatch (2) rmatch 3 4

G2 mass 2 { } nonaxisymmetric velocity 6 L ⇠ c6 size of source 2 { } { }

1000 0100 (3) 2 00103 6 00017 4 5

2M 1+ r 000 1 0 1 2M 00 2 r 3 (4) 00 r2 0 6 000 r2 sin2 ✓7 6 7 4 5 IMR16 ⇡CONSISTENCYG TEST: h¯µ⌫ = ( matter contributions + O(h¯2) contributions ), (5) ⇤FORMULATION c4 { } { } d2 PHYSICAL REVIEW LETTERS week ending +(!2 V ) =0, PRL 116, 221101 (2016) (6) 3 JUNE 2016 dr2 ⇤

r Infer finalr mass:= r +2 Mandlog spin 1from, low- (7) ⇤ 2M and high-frequency ⇣parts ⌘of the signal — roughly2M inspirall(l + 1) and6M V (r)= 1 (8) r r2 r3 merger-ringdown  parts

+ (9) Check consistency··· of these FIG. 2. MAP estimate and 90% credible regions for (upper inferences and combine20 together + O(v ) panel)illustration the waveform and from (lower(10) panel) Tests the GWof frequencyGeneral of FIG. 3. Frequency regions of the parametrized waveform model GW150914 as estimated by the LALINFERENCE analysis [3]. The as defined in the text and in Ref. [41]. The plot shows the absolute results using fractional parameters: solid linesRelativity in each panel indicatewith the GW150914 most-probable waveform, value of the frequency-domain amplitude of the most-probable from GW150914 [3] and its GW frequency. We mark with a waveform from GW150914 [3]. The inspiral region (cyan) from LVC PRL (2016)end insp 20 to ∼55 Hz corresponds to the early- and late-inspiral regimes. insp post-insp insp post-insp vertical line the instantaneous frequency fGW 132 Hz, Mf Mf Mf af af af which is used in the IMR consistency test to delineate¼ the The intermediate region (red) goes from ∼55 to ∼130 Hz. := 2 insp post-insp , := 2 insp post-insp boundary between the frequency-domain(11) inspiral and postinspiral Finally, the merger-ringdown region (orange) goes from M¯ f a¯f Mf + Mf af + af parts (see Fig. 3 below for a representation of the most-probable ∼130 Hz to the end of the waveform. waveform’s amplitude in frequency domain). 10 expected across those bands. Large matched-filter SNR Test introduced in Ghosh et al. PRD (2016) & CQG (2018), building onend earlier insp proposals for LISA 2 20 Hz to fGW , and we estimate the posterior dis- values which are accompanied by a large χ statistic are very tributions of the binary’s component masses and spins likely due either to noise glitches or to a mismatch between using this “inspiral” (low-frequency) part of the observed the signal and the model matched-filter waveform. signal, using the nested-sampling algorithm in the Conversely, reduced-χ2 values near unity indicate that the LALINFERENCE software library [52]. We then use for- data are consistent with waveform plus the expected detector mulas obtained from NR simulations to compute posterior noise. Thus, large χ2 values are a warning that some parts of distributions of the remnant’s mass and spin. Next, we the waveform are a much worse fit than others, and thus the obtain the complementary “postinspiral” (high-frequency) candidates may result from instrument glitches that are very signal, which is dominated by the contribution from the loud, but they do not resemble binary-inspiral signals. merger and ringdown stages, by restricting the frequency- However, χ2 tests are performed by comparing the data domain representation of the waveforms to extend between with a single theoretical waveform, while in this case we end insp fGW and 1024 Hz. Again, we derive the posterior allow the inspiral and postinspiral partial waveforms to distributions of the component masses and spins, and select different physical parameters. Thus, this test should be (by way of NR-derived formulas) of the mass and spin sensitive to subtler deviations from the predictions of GR. of the final compact object. We note that the MAP wave- In Fig. 4 we summarize our findings. The top panel form has an expected SNRdet ∼ 19.5 if we truncate its shows the posterior distributions of Mf and af estimated frequency-domain representation to have support between from the inspiral and postinspiral signals, and from the 20 and 132 Hz, and ∼16 if we truncate it to have support entire inspiral-merger-ringdown waveform. The plot con- between 132 and 1024 Hz. Finally, we compare these two firms the expected behavior: the inspiral and postinspiral estimates of the final Mf and dimensionless spin af and 90% confidence regions (defined by the isoprobability compare them also against the estimate performed using contours that enclose 90% of the posterior) have a full inspiral-merger-ringdown waveforms. In all cases, we significant region of overlap. As a sanity check (which, average the posteriors obtained with the EOBNR and strictly speaking, is not part of the test of GR that is being IMRPHENOM waveform models, following the procedure performed), we also produced the 90% confidence region outlined in Ref. [3]. Technical details about the imple- computed with the full inspiral-merger-ringdown wave- mentation of this test can be found in Ref. [60]. form; it lies comfortably within this overlap. We have This test is similar in spirit to the χ2 GW search statistic verified that these conclusions are not affected by the [2,61], which divides the model waveform into frequency specific formula [40,59,62] used to predict Mf and af, or bands and checks to see that the SNR accumulates as by the choice of fend insp within 50 Hz. GW Æ

221101-4 4 IMR CONSISTENCY TEST: EVENTS TABLE I. The GW events considered in this paper, separated by observing run. The first block of columns gives the names of the events and lists some of their relevant properties obtained using GR waveforms (luminosity distance DL, source frame total mass Mtot and final mass Mf, and dimensionless final spin af). The next block of columns gives the significance, measured by the false-alarm-rate (FAR), with which each event was detected by each of the three searches employed, as well as the matched filter signal-to-noise ratio from the stochastic sampling analyses with GR waveforms. A dash indicates that an event was not identified by a search. The parameters and SNR values give the medians Can only apply thisand test 90% credible to intervals.events All the with events except sufficient for GW151226 SNR and GW170729 in both are consistent with a binary of nonspinning black holes (when analyzed assuming GR). See [14] for more details about all the events.a The last block of columns indicates which GR tests are performed on a high- and low-frequencygiven event: RT = residuals parts test (Sec.(high-massVA); IMR = inspiral-merger-ringdown systems only) consistency test (Sec. VB); PI & PPI = parameterized tests of GW generation for inspiral and post-inspiral phases (Sec. VI); MDR = modified GW dispersion relation (Sec. VII). The events with bold names are used to obtain the combined results for each test. Properties FAR GR tests performed Event SNR DL Mtot Mf af PyCBC GstLAL cWB RT IMR PI PPI MDR 1 1 1 [Mpc] [M ][M ] [yr ] [yr ] [yr ] +150 +3.7 +3.3 +0.05 5 7 4 +0.1 3333 3 GW150914 430 170 66.2 3.3 63.1 3.0 0.69 0.04 < 1.5 10 < 1.0 10 < 1.6 10 25.3 0.2 +550 +10.6 +10.7 +0.13 ⇥ ⇥ 3 ⇥ +0.3 GW151012 1060 37.3 35.7 0.67 0.17 7.9 10 – 9.2 3 ––33 480 3.9 3.8 0.11 ⇥ 0.4 +180 +6.2 +6.4 +0.07 5 7 +0.2 3 3 3 GW151226 440 190 21.5 1.5 20.5 1.5 0.74 0.05 < 1.7Events10 < 1suitable.0 10 0.02 12.4 0.3 – – ⇥ ⇥ +440 +5.3 +5.2 +0.08 5 7 4 +0.2 3333 3 GW170104 960 420 51.3 4.2 49.1 4.0 0.66 0.11 < 1.4 for10 the< 1.0 IMR10 2.9 10 14.0 0.3 +120 +3.1 +3.2 +0.04 ⇥ 4 ⇥ 7 ⇥ 4 +0.2 GW170608 320 18.6 17.8 0.69 < 3.1 10 < 1.0 10 1.4 10 15.6 3 – 33 3 110 0.7 0.7 0.04 ⇥consistency⇥ ⇥ 0.3 +1380 +15.6 +14.6 +0.07 +0.4 33 33 GW170729 2760 1340 85.2 11.1 80.3 10.2 0.81 0.13 1.4 0.18 0.02 10.8 0.5 – +320 +5.4 +5.2 +0.08 4 test 7 +0.2 GW170809 990 59.2 56.4 0.70 1.4 10 < 1.0 10 – 12.7 33– 33 380 3.9 3.7 0.09 ⇥ ⇥ 0.3 +160 +3.4 +3.2 +0.07 5 7 4 +0.3 3333 3 GW170814 580 210 56.1 2.7 53.4 2.4 0.72 0.05 < 1.2 10 < 1.0 10 < 2.1 10 17.8 0.3 +430 +5.1 +4.8 +0.07 ⇥ ⇥ 5 ⇥ +0.3 GW170818 1020 62.5 59.8 0.67 – 4.2 10 – 11.9 33– 33 360 4.0 3.8 0.08 ⇥ 0.4 +840 +9.9 +9.4 +0.08 5 7 3 +0.2 33 33 GW170823 1850 840 68.9 7.1 65.6 6.6 0.71 0.10 < 3.3 10 < 1.0 10 2.1 10 12.1 0.3 – ⇥ ⇥ ⇥ a The parameters given in this table di↵er slightly from those in v2 of the arXiv version of [14] since they correct for a small issue in the application of priors for spin parameters. 11

285 respect to the detector network. The intrinsic parameters for 309 aligned-spin model. We use IMRPhenomPv2 to obtain all the 286 circularized black-hole binaries in GR are the two masses mi of 310 main results given in this paper, and use SEOBNRv4 to check 287 the black holes and the two spin vectors S~ i defining the rotation 311 the robustness of these results, whenever possible. When we 288 of each black hole, where i 1, 2 labels the two black holes. 312 use IMRPhenomPv2, we impose a prior m /m 18 on the 2 { } 1 2  289 We assume that the binary has negligible orbital eccentricity, 313 mass ratio, as the waveform family is not calibrated against 290 as is expected to be the case when the binary enters the band of 314 numerical relativity simulations for m1/m2 > 18. We do not 291 ground-based detectors [45, 46] (except in some more extreme 315 impose a similar prior when using SEOBNRv4, since it in- 4 292 formation scenarios, e.g., [54–57]). The extrinsic parameters 316 cludes information about the extreme mass ratio limit. Neither 293 comprise four parameters that specify the space-time location 317 of these waveform models includes the full spin dynamics 294 of the binary black-hole, namely the sky location (right ascen- 318 (which requires 6 spin parameters). Fully precessing waveform 295 sion and declination), the luminosity distance, and the time of 319 models have been recently developed [21, 60–62] and will be 296 coalescence. In addition, there are three extrinsic parameters 320 used in future applications of these tests.

297 that determine the orientation of the binary with respect to 321 The waveform models used in this paper do not include the 298 Earth, namely the inclination angle of the orbit with respect 322 e↵ects of subdominant (non-quadrupole) modes, which are 299 to the observer, the polarization angle, and the orbital phase at 323 expected to be small for comparable-mass binaries [63, 64]. 300 coalescence. 324 The first generation of binary black hole waveform models 301 We employ two waveform families that model binary black 325 including spin and higher order modes has recently been de- 302 v holes in GR: the e↵ective-one-body based SEOBNR 4 [18] 326 veloped [62, 65–67]. Preliminary results in [14], using NR 303 waveform family that assumes non-precessing spins for the 327 simulations supplemented by NR surrogate waveforms, indi- 304 black holes (we use the frequency domain reduced order 328 cate that the higher mode content of the GW signals detected 305 v model SEOBNR 4 ROM for reasons of computational e- 329 by Advanced LIGO and Virgo is weak enough that models 306 he ciency), and the phenomenological waveform family IMRP - 330 without the e↵ect of subdominant modes do not introduce sub- 307 nom v P 2 [19, 58, 59] that models the e↵ects of precessing spins 331 stantial biases in the intrinsic parameters of the binary. For 308 using two e↵ective parameters by twisting up the underlying 332 unequal-mass binaries, the e↵ect of the non-quadrupole modes 333 is more pronounced [68], particularly when the binary’s ori- 334 entation is close to edge-on. In these cases, the presence of 4 These scenarios could occur often enough, compared to the expected rate 335 non-quadrupole modes can show up as a deviation from GR of detections, that the inclusion of eccentricity in waveform models is a 336 when using waveforms that only include the quadrupole modes, necessity for tests of GR in future observing runs; see, e.g., [47–53] for 337 as was shown in [69]. Applications of tests of GR with the new recent work on developing such waveform models. 338 waveform models that include non-quadrupole modes will be IMR CONSISTENCY TEST: RESULTS 7

8 501 inference of the full signal (see Table III for values of fc). The 1.5 1.0 0.5 0.0 0.5 1.0 1.5 502 binary’s parameters are then estimated independently from the 6 GW150914 503 lowAll (high) events frequency consistent parts of the data with by restricting GR the noise- 5 GW170104 ) f GW170729 504 weighted integral in the likelihood calculation to frequencies ¯ expectations—GR recovered M 4

/ GW170809 505 below (above) this frequency cuto↵ f . For each of these inde-

f c 3 M GW170814 506 pendentat < estimates 80% credible of the source parameters,level we make use of fits ( 2 GW170818 P 507 to numerical-relativity simulations given in [77–79] to infer the 1 GW170823 (largest for GW170823). 2 508 M a = c S~ / GM 1.0 1.0 mass f and dimensionless spin magnitude f f ( f ) 9 | | 509 of the remnant black hole. If the data are consistent with GR, prior 510 theseSEOBNRv4_ROM two independent estimates results have to be the consistent with 0.5 0.5 511 each other [38, 75]. Because this consistency test ultimately 512 comparessame, between qualitatively the inspiral and the post-inspiral results, 513 f posteriors of both parts must be informative. In the case of ¯ a / f 0.0 0.0 514 low-mass binaries, the SNR in the part f > fc is insucient a 515 toMultimodal perform this test, soresults that we only for analyze seven events as 516 indicated in Tables I and III. 0.5 0.5 517 GW170814In order to quantify and the consistencyGW170823 of the two di↵erent 518 estimateslikely ofdue the final to blacknoise—also hole’s mass and seen spin we define 519 two dimensionless quantities that quantify the fractional di↵er- insp post-insp insp 1.0 1.0 520 encefor between injections them: M fin/M¯ frealB 2( MnoiseM )/(M + 1.5 1.0 0.5 0.0 0.5 1.0 1.5 1 2 3 4 5 f f f post-insp insp post-insp insp post-insp ¯ / 521 M ) and af/a¯f B 2(a a )/(a + a ), Mf/Mf P( af a¯f) f f f f f 522 where the superscripts indicate the estimates of 12 the mass 523 and spin from the inspiral and post-inspiral parts of the sig- 10 FIG. 2. Results of the inspiral-merger-ringdown consistency test for 524 nal. The posteriors of these dimensionless parameters, es- the selected BBH events (see Table I). The main panel shows 90% 525 timated from di↵erent events, are shown in Fig. 2. For credible regions of the posterior distributions of (Mf /M¯ f , af /a¯f ), 526 all events, the posteriors are consistent with the GR value with the cross marking the expected value for GR. The side panels 527 (Mf/M¯ f = 0, af/a¯f = 0). The fraction of the posterior show the marginalized posteriors for Mf /M¯ f and af /a¯f . The thin 528 enclosed by the isoprobability contour that passes through black dashed curve represents the prior distribution, and the grey 529 the GR value (i.e., the GR quantile) for each event is shown shaded areas correspond to the combined posteriors from the five 530 in Table III. Figure 2 also shows the posteriors obtained by most significant events (as outlined in Sec. III and Table I). 531 combining all the events that pass the stronger significance 1 532 threshold FAR < (1000 yr) , as outlined in Sec. III (see the 533 same section for a discussion of caveats). TABLE III. Results from the inspiral-merger-ringdown consistency 534 The parameter estimation is performed employing uniform ↵ test for selected binary black hole events. fc denotes the cuto fre- 535 priors in component masses and spin magnitudes and isotropic quency used to demarcate the division between the inspiral and post- 536 priors in spin directions [14]. This will introduce a non-flat inspiral regimes; ⇢IMR, ⇢insp, and ⇢post insp are the median values of 537 prior in the deviation parameters M /M¯ and a /a¯ , which the SNR in the full signal, the inspiral part, and the post-inspiral part, f f f f 538 is shown as a thin, dashed contour in Fig. 2. Posteriors are respectively; and the GR quantile denotes the fraction of the posterior enclosed by the isoprobability contour that passes through the GR 539 estimated employing the precessing spin phenomenological value, with smaller values indicating better consistency with GR. 540 waveform family IMRPhenomPv2. To assess the systematic 541 errors due to imperfect waveform modeling, we also estimate Event fc [Hz] ⇢IMR ⇢insp ⇢post insp GR quantile [%] 542 the posteriors using the e↵ective-one-body based waveform GW150914 132 25.3 19.4 16.1 55.5 543 family SEOBNRv4 that models binary black holes with non- GW170104 143 13.7 10.9 8.5 24.4 544 precessing spins. There is no qualitative di↵erence between GW170729 91 10.7 8.6 6.9 10.4 GW170809 136 12.7 10.6 7.1 14.7 GW170814 161 16.8 15.3 7.2 7.8 8 GW170818 128 12.0 9.3 7.2 25.5 The frequency fc was determined using preliminary parameter inference GW170823 102 11.9 7.9 8.5 80.4 results, so the values in Table III are slightly di↵erent than those that would be obtained using the posterior samples in GWTC-1 [9]. However, the test is robust against small changes in the cuto↵ frequency [38]. 9 As in [6], we average the Mf , af posteriors obtained by di↵erent fits [77–79] 494 part (to a good approximation, produced by the post-inspiral after augmenting the fitting formulae for aligned-spin binaries by adding the contribution from in-plane spins [80]. However, unlike in [6, 80], we do 495 stages). The cuto↵ frequency fc between the two regimes is not evolve the spins before applying the fits, due to technical reasons. 496 chosen as the frequency of the innermost stable circular or- 10 For black hole binaries with comparable masses and moderate spins, as we 497 bit of a Kerr black hole [76], with mass and dimensionless consider here, the remnant black hole is expected to have af & 0.5; see, e.g., 498 spin equal to the median values of the posterior distribution [77–79] for fitting formulae derived from numerical simulations, or Table I 499 of the remnant’s mass and spin. This determination of fc is for values of the remnant’s spins obtained from GW events. Hence, af /a¯f is expected to yield finite values. 500 performed separately for each event and based on parameter 8

545 the results derived using the two di↵erent waveforms families 600 as well as the tightest combined constraints obtained to date 546 (see Sec. 2 of the Appendix). 601 by combining information from all the significant binary black 547 We see additional peaks in the posteriors estimated from 602 holes events observed so far, as described in Sec. III. 548 GW170814 and GW170823. Detailed follow-up investigations 603 The frequency-domain GW phase evolution ( f ) in the 549 did not show any evidence of the presence of a coherent signal 604 early-inspiral stage of IMRPhenomPv2 is described by a PN 550 in multiple detectors that di↵ers from the GR prediction. The 605 expansion, augmented with higher-order phenomenological co- 551 second peak in GW170814 is introduced by the posterior of 606 ecients. The PN phase evolution is analytically expressed in post-insp 607 closed form by employing the stationary phase approximation. 552 Mf , while the extra peak in GW170823 is introduced by insp 608 The late-inspiral and post-inspiral (intermediate and merger- 553 the posterior of Mf . Injection studies in real data around 609 ringdown) stages are described by phenomenological analytical 554 the time of these events, using simulated GR waveforms with 610 expressions. In all cases, the phenomenological coecients 555 parameters consistent with GW170814 and GW170823, sug- PARAMETERIZED TEST OF GW 611 are calibrated with data from NR simulations of mass-ratios as 556 gest that such secondary peaks occur for 10% of injections. GENERATION: PHYSICALFORMULATION REVIEW LETTERS week ending ⇠ 612 asymmetric as 1 : 18 and of dimensionlessPRL 116, 221101 (2016) spin-magnitudes up 3 JUNE 2016 557 Features in the posteriors of GW170814 and GW170823 are 613 to 0.99, as well as the inspiralAnalytical portion predictions of (uncalibrated) of GR in the EOB 558 thus consistent with expected noise fluctuations. 11 614 waveforms. The transitioninspiral frequency given by post-Newtonianfrom inspiral to inter- 3 615 mediate regime is given bycoefficients. the condition GMf/c = 0.018, 616 with M the total mass of the binary in the detector frame, since 559 VI. PARAMETERIZED TESTS OF GRAVITATIONAL Introduce parameterized deviations in 617 this is the lowest frequency above which this model was cal- 560 WAVE GENERATION PN phasing coefficients, as well as 618 ibrated with NR data [19].phenomenological Deviations from coefficients GR in fit allto NR three 619 simulationsFIG. 2.at MAP higher estimate andfrequencies: 90% credible regions for (upper illustration from Tests of General stages are expressed by means of relativepanel) the waveform shifts and (lower panel)pˆ thei in GW frequency the corre- of FIG. 3. Frequency regions of the parametrized waveform model as defined in the textRelativity and in Ref. with[41] .GW150914 The plot shows, the absolute 561 A deviation from GR could manifest itself as a modification GW150914 as estimated by the LALINFERENCE analysis [3]. The 620 sponding waveform coecients: psolidi lines in each(1 panel+ indicatepˆ thei) most-probablepi, which waveform arevalue of the frequency-domainLVC PRL amplitude (2016) of the most-probable from GW150914 [3] and its GW frequency. We mark with a waveform from GW150914 [3]. The inspiral region (cyan) from 562 SEBASTIAN KHAN et al. end insp PHYSICAL REVIEW D 93, 044007 (2016) of the dynamics of two orbiting compact objects, and in par- vertical! line the instantaneous frequency f 132 Hz, 20 to ∼55 Hz correspondsa few to thePN early- coefficients and late-inspiral regimes. 621 GW ¼ The intermediate region (red) goes from 55 to 130 Hz. used as additional free parameterswhich in is used our in the extended IMR consistency test to waveform delineate the 3715 55η ∼ ∼ Vary oneThe coefficient expansion coefficients at area thentime—sufficient given by Finally, the merger-ringdown region (orange) goes from 563 boundary between the frequency-domain inspiral and postinspiral φ2 ; B8 ticular, the evolution of the orbital (and hence, GW) phase. In ∼130 Hz to the¼ end756 of theþ waveform.9 ð Þ 622 to pick upparts deviations (see Fig. 3 below for afromφ representation1; GR of the most-probableB6 models. waveform’s amplitude in frequency0 domain). 564 henom v ¼ ð Þ an analytical waveform model like IMRP P 2, the de- expected across those bands.113δχ Largea 113 matched-filter76η SNR 623 φ1 0; B7 φ3 −16π − χs; B9 We denote the testingCurrent parameters implementation20 Hz corresponding of to testfend described insp, and we in¼ estimate Mediam the to posterioret PN al. PRD dis-ð phase Þ(2018)values which are¼ accompaniedþ 3 by aþ large3χ2 statistic3 are very 13ð Þ GW   565 tails of the GW phase evolution are controlled by coecients tributions of the binary’s component masses and spins likely due either to noise glitches or to a mismatch between 624 coecients by 'ˆi, where i indicatesusing this the“inspiral power” (low-frequency) of partv/ of thec observedbeyondthe signal and the model matched-filter waveform. signal, using the nested-sampling algorithm in the Conversely, reduced-χ2 values near unity indicate that the 566 that are either analytically calculated or determined by fits to 2 LALINFERENCE15293365software27145 libraryη [52]3085.η We then405 use for- data2 are405 consistent with waveform405 5η plus2 the expected detector 625 leading (Newtonian or 0PN) orderφ4 in ( f ). The frequency− 200η χa − δχaχs 2 − χs ; B10 mulas obtained¼ 508032 from NRþ simulations504 þ 72 to computeþ posterior8 þ noise. Thus,4 largeþχ values8 areþ a2 warning that some partsð of Þ 567 numerical-relativity (NR) simulations, always under the as- distributions of the remnant’s mass and(i spin.5)/ Next,3 we the waveform are a much worse fit than others, and thus the 626 dependence of the correspondingobtain phase the complementary term“postinspiral38645 is πf 65” (high-frequency)πη .732985 In thecandidates140η may result732985 from instrument24260η glitches340η2 that are very φ 1 log πMf − δ − − χ − χ ; B11 568 sumption that GR is the underlying theory. These coecients signal,5 which¼½þ is dominatedð ފ by756 the contribution9 þ from2268 the loud,9 buta theyþ do2268 not resembleþ 81 binary-inspiralþ 9 s signals.ð Þ 627 parametrized model, i varies frommerger 0 and to ringdown 7, including stages, by restricting the the frequency- termsHowever, χ2 tests are performed by comparing  the data domain representation11583231236531 of the waveforms6848γ to640 extendπ2 between15737765635with a single2255 theoreticalπ2 76055 waveform,η2 127825 while inη3 this case we 569 end insp E have a functional dependence on the intrinsic parameters of the f φ6 and 1024 Hz. Again,− we derive− the posterior− allow the inspiral andη postinspiral− partial waveforms to GW ¼ 4694215680 21 3 þ 3048192 þ 12 þ 1728 1296 628 with logarithmic dependence atdistributions 2.5PN of the and component 3PN. masses and The spins, and non-select different physical parameters. Thus, this test should be 570 (by way of6848 NR-derived formulas)2270 of the mass and2270 spinπ sensitive to subtler deviations from the predictions of GR. binary that is specific to GR (masses and spins for binary black − log 64πMf πδχa − 520πη χs; B12 629 logarithmic term at 2.5PN (i.e., i of= the final5) compact cannot63 object.ð WeÞþ be note3 that constrained, theþ MAP wave-3 In Fig. 4 we summarize our findings. The top panelð Þ  shows the posterior distributions of M and a estimated 571 form has an expected SNRdet ∼ 19.5 if we truncate its f f holes). In this section we investigate deviations from the GR frequency-domain77096675 representationπ 378515πη to have74045 supportπη2 between25150083775from the inspiral26804935 andη postinspiral1985η2 signals, and from the 630 φ − δ − − χ because of its degeneracy with a constant20 and7 132¼ Hz,254016 and reference∼16þ if1512 we truncate it756 tophase haveþ support (e.g.,3048192entire inspiral-merger-ringdownþ 6048 48 waveform.a The plot con-  firms the expected behavior: the inspiral and postinspiral 572 binary dynamics by introducing shifts in each of the individ- between 132 and 1024 Hz. Finally, we compare these two 2 3 estimates of the25150083775 final M and dimensionless10566655595 spinη a1042165and η90%5345 confidenceη regions (defined by the isoprobability 631 the phase at coalescence). These coecients− f were introduced− f χs: B13 3048192 762048 3024 36 573 compare themþ also against theþ estimate performed using contoursþ that enclose 90% of the posterior) haveð a Þ ual GW phase coecients of IMRPhenomPv2. We then treat  significant region of overlap. As a sanity check (which, 632 full inspiral-merger-ringdown waveforms. In all cases, we in their current form in Eq. (19) ofaverage [89 the posteriors]. In obtained addition, with the EOBNR we and alsostrictly speaking, is not part of the test of GR that is being 574 those shifts as additional unconstrained GR-violating parame- IMRPAs discussedHENOM waveform in Secs. VI models, B and followingIV in Paper the 1, procedure our inspiral performed), we also produced the 90% confidence region 2 633 test for i = 2, representing an e↵amplitudeoutlinedective in model Ref. [3] is based.1PN Technical on a reexpanded details term, about PN the which amplitude. imple- computed is 27312085 with the full1975055 inspiral-merger-ringdownη 105271η wave- A4 − − 575 Thementation expansion of this coefficients test can be of found Eq. (29) in Ref.are[60] given. by form;¼ it lies8128512 comfortably338688 withinþ this24192 overlap. We have ters, which we measure in addition to the standard parameters This test is similar in spirit to the χ2 GW search statistic verified that these conclusions are not affected by the 634 81 2 81 81 17η 2 motivated below. The full set of inspiral[2,61], which divides parameters the model waveform into are frequency thusspecific formula− [40,59,62]8η χa −usedδχ toa predictχs M− f and af, orχs ; A0 1; B14 þ 32 þ 16 þ 32 þ 8 576 describing the binary. bands and checks¼ to see that the SNR accumulatesð as Þ by the choice of fend insp within 50 Hz.   GW Æ B18 A1 0; B15 ð Þ ¼ ð Þ 577 The early inspiral of compact binaries is well modeled by 221101-4 'ˆ 2, 'ˆ0, 'ˆ1, 'ˆ2, 'ˆ3, 'ˆ4, 'ˆ5l, 'ˆ3236, 451'ˆη 6l, 'ˆ7 . 85π 85πη 285197 1579η A2 − ; B16 A − δ − χ 578 { ¼ 224 þ 168 } ð Þ 5 ¼ 64 þ 16 þ 16128 4032 a the post-Newtonian (PN) approximation [81–84] to GR, which   2 27δχa 27 11η 285197 15317η 2227η A − χ ; B17 − − χs; B19 579 is based on the expansion of the orbital quantities in terms of 635 Since the 1PN term and the 0.5PN term3 ¼ are8 þ absent8 6 s in theð Þ GR þ 16128 672 1008 ð Þ     580 a small velocity parameter v/c. For a given set of intrinsic 636 phasing, we parametrize 'ˆ and 'ˆ as absolute deviations, 2 1 177520268561 545384828789 205π2 3248849057η2 34473079η3 A − − η − 6 ¼ 8583708672 þ 5007163392 48 178827264 þ 6386688 581 parameters, coecients for the di↵erent orders in v/c in the PN 637 with a pre-factor equal to the 0PN coecient.   1614569 1873643η 2167η2 31π 7πη 1614569 61391η 57451η2 − χ2 − χ − χ2 582 þ 64512 16128 þ 42 a þ 12 3 s þ 64512 1344 þ 4032 s series are uniquely determined. A consistency test of GR using 638 The 1PN term of 'ˆ2 can be interpreted as arising  from   31π 1614569 165961η δχ − χ : B20 583 þ a 12 þ 32256 2688 s ð Þ measurements of the inspiral PN phase coecients was first 639 the emission of dipolar radiation. For binary  black holes,  this

584 proposed in [85–87], while a generalized parametrization was 640 could occur in, e.g., alternative theories of gravityAPPENDIX C: where PHENOMENOLOGICAL an COEFFICIENTS The values of the coefficients for the mapping functions given in Eq. (31) are shown in Table V. These values are 585 motivated in [88]. Bayesian implementations based on such 641 additional scalar charge is sourced by terms related to curva- calculated under the parametrization (η, χPN). 586 parametrized methods were presented and tested in [39, 89– 642 ture [95, 96]. At leading order, this introduces a deviation in 587 91] and were also extended to the post-inspiral part of the 643 the 1PN coecient of the waveform [97, 98]. This e044007-24↵ec- 588 gravitational-wave signal [92, 93]. These ideas were applied 644 tively introduces a term in the inspiral GW phase, varying with 7/3 589 to the first GW observation, GW150914 [10], yielding the first 645 frequency as f , while the gravitational flux is modified as 2 2 590 bounds on higher-order PN coecients [4]. Since then, the 646 GR GR(1 + Bc /v ). The first bound on 'ˆ 2 was pub- F ! F 591 constraints have been revised with the binary black hole events 647 lished in [8]. The higher-order terms in the above expansion 592 that followed, GW151226 in O1 [5] and GW170104 in O2 [6]. 648 also lead to a modification in the higher-order PN coecients. 593 More recently, the first such constraints from a binary neutron 649 Unlike the case of GW170817 (which we study separately 594 star merger were placed with the detection of GW170817 [8]. 650 in [8]), where the higher-order terms in the expansion of the 595 Bounds on parametrized violations of GR from GW detections 596 have been mapped, to leading order, to constraints on specific 597 alternative theories of gravity (see, e.g., [94]). In this paper, we 11 598 present individual constraints on parametrized deviations from This frequency is di↵erent than the cuto↵ frequency used in the inspiral- merger-ringdown consistency test, as was briefly mentioned in Sec. III. 599 GR for each of the GW sources in O1 and O2 listed in Table I, PARAMETERIZED TEST OF GW 4

TABLE I. The GW events considered in this paper, separated by observing run. The first block of columns gives the names of the events and GENERATION:lists some of their EVENTS relevant properties obtained using GR waveforms (luminosity distance DL, source frame total mass Mtot and final mass Mf, and dimensionless final spin af). The next block of columns gives the significance, measured by the false-alarm-rate (FAR), with which each event was detected by each of the three searches employed, as well as the matched filter signal-to-noise ratio from the stochastic sampling analyses with GR waveforms. A dash indicates that an event was not identified by a search. The parameters and SNR values give the medians Can only test the inspiral/post-inspiraland 90% credible intervals. All the events coefficients except for GW151226 on and systems GW170729 are consistent with with a binary of nonspinning black holes (when analyzed assuming GR). See [14] for more details about all the events.a The last block of columns indicates which GR tests are performed on a enoughgiven SNR event: RT in= residualsthe appropriate test (Sec. VA); IMR = inspiral-merger-ringdown bands consistency test (Sec. VB); PI & PPI = parameterized tests of GW generation for inspiral and post-inspiral phases (Sec. VI); MDR = modified GW dispersion relation (Sec. VII). The events with bold names are used to obtain the combined results for each test. Properties FAR GR tests performed Event SNR DL Mtot Mf af PyCBC GstLAL cWB RT IMR PI PPI MDR 1 1 1 [Mpc] [M ][M ] [yr ] [yr ] [yr ] inspiral +150 +3.7 +3.3 +0.05 5 7 4 +0.1 3333 3 GW150914 430 170 66.2 3.3 63.1 3.0 0.69 0.04 < 1.5 10 < 1.0 10 < 1.6 10 25.3 0.2 +550 +10.6 +10.7 +0.13 ⇥ ⇥ only3 ⇥ +0.3 GW151012 1060 37.3 35.7 0.67 0.17 7.9 10 – 9.2 3 ––33 480 3.9 3.8 0.11 ⇥ 0.4 +180 +6.2 +6.4 +0.07 5 7 +0.2 3 3 3 GW151226 440 190 21.5 1.5 20.5 1.5 0.74 0.05 < 1.7 10 < 1.0 10 0.02 12.4 0.3 – – ⇥ ⇥ +440 . +5.3 . +5.2 . +0.08 < . 5 < . 7 . 4 . +0.2 3333 3 GW170104 960 420 51 3 4.2 49 1 4.0 0 66 0.11 1 4 10 1 0post-inspiral10 2 9 10 14 0 0.3 +120 +3.1 +3.2 +0.04 ⇥ 4 ⇥ 7 ⇥ 4 +0.2 GW170608 320 18.6 17.8 0.69 < 3.1 10 < 1.0 10 1.4 10 15.6 3 – 33 3 110 0.7 0.7 0.04 ⇥ ⇥ ⇥ 0.3 +1380 +15.6 +14.6 +0.07 only +0.4 33 33 GW170729 2760 1340 85.2 11.1 80.3 10.2 0.81 0.13 1.4 0.18 0.02 10.8 0.5 – +320 +5.4 +5.2 +0.08 4 7 +0.2 GW170809 990 59.2 56.4 0.70 1.4 10 < 1.0 10 – 12.7 33– 33 380 3.9 3.7 0.09 ⇥ ⇥ 0.3 +160 +3.4 +3.2 +0.07 5 7 4 +0.3 3333 3 GW170814 580 210 56.1 2.7 53.4 2.4 0.72 0.05 < 1.2 10 < 1.0 10 < 2.1 10 17.8 0.3 +430 +5.1 +4.8 +0.07 ⇥ ⇥ 5 ⇥ +0.3 GW170818 1020 62.5 59.8 0.67 – 4.2 10both – 11.9 33– 33 360 4.0 3.8 0.08 ⇥ 0.4 +840 +9.9 +9.4 +0.08 5 7 3 +0.2 33 33 GW170823 1850 840 68.9 7.1 65.6 6.6 0.71 0.10 < 3.3 10 < 1.0 10 2.1 10 12.1 0.3 – ⇥ ⇥ ⇥ a The parameters given in this table di↵er slightly from those in v2 of the arXiv version of [14] since they correct for a small issue in the application of priors for spin parameters. 14

285 respect to the detector network. The intrinsic parameters for 309 aligned-spin model. We use IMRPhenomPv2 to obtain all the 286 circularized black-hole binaries in GR are the two masses mi of 310 main results given in this paper, and use SEOBNRv4 to check 287 the black holes and the two spin vectors S~ i defining the rotation 311 the robustness of these results, whenever possible. When we 288 of each black hole, where i 1, 2 labels the two black holes. 312 use IMRPhenomPv2, we impose a prior m /m 18 on the 2 { } 1 2  289 We assume that the binary has negligible orbital eccentricity, 313 mass ratio, as the waveform family is not calibrated against 290 as is expected to be the case when the binary enters the band of 314 numerical relativity simulations for m1/m2 > 18. We do not 291 ground-based detectors [45, 46] (except in some more extreme 315 impose a similar prior when using SEOBNRv4, since it in- 4 292 formation scenarios, e.g., [54–57]). The extrinsic parameters 316 cludes information about the extreme mass ratio limit. Neither 293 comprise four parameters that specify the space-time location 317 of these waveform models includes the full spin dynamics 294 of the binary black-hole, namely the sky location (right ascen- 318 (which requires 6 spin parameters). Fully precessing waveform 295 sion and declination), the luminosity distance, and the time of 319 models have been recently developed [21, 60–62] and will be 296 coalescence. In addition, there are three extrinsic parameters 320 used in future applications of these tests.

297 that determine the orientation of the binary with respect to 321 The waveform models used in this paper do not include the 298 Earth, namely the inclination angle of the orbit with respect 322 e↵ects of subdominant (non-quadrupole) modes, which are 299 to the observer, the polarization angle, and the orbital phase at 323 expected to be small for comparable-mass binaries [63, 64]. 300 coalescence. 324 The first generation of binary black hole waveform models 301 We employ two waveform families that model binary black 325 including spin and higher order modes has recently been de- 302 v holes in GR: the e↵ective-one-body based SEOBNR 4 [18] 326 veloped [62, 65–67]. Preliminary results in [14], using NR 303 waveform family that assumes non-precessing spins for the 327 simulations supplemented by NR surrogate waveforms, indi- 304 black holes (we use the frequency domain reduced order 328 cate that the higher mode content of the GW signals detected 305 v model SEOBNR 4 ROM for reasons of computational e- 329 by Advanced LIGO and Virgo is weak enough that models 306 he ciency), and the phenomenological waveform family IMRP - 330 without the e↵ect of subdominant modes do not introduce sub- 307 nom v P 2 [19, 58, 59] that models the e↵ects of precessing spins 331 stantial biases in the intrinsic parameters of the binary. For 308 using two e↵ective parameters by twisting up the underlying 332 unequal-mass binaries, the e↵ect of the non-quadrupole modes 333 is more pronounced [68], particularly when the binary’s ori- 334 entation is close to edge-on. In these cases, the presence of 4 These scenarios could occur often enough, compared to the expected rate 335 non-quadrupole modes can show up as a deviation from GR of detections, that the inclusion of eccentricity in waveform models is a 336 when using waveforms that only include the quadrupole modes, necessity for tests of GR in future observing runs; see, e.g., [47–53] for 337 as was shown in [69]. Applications of tests of GR with the new recent work on developing such waveform models. 338 waveform models that include non-quadrupole modes will be PARAMETERIZED TEST OF GW GENERATION:

PN COEFFICIENT CONSTRAINTS 10

2 2 2 759 dispersion relation E = p c of GWs in GR, giving

2 2 2 ↵ ↵ E = p c + A↵ p c . (2) 101 Improve previous combined constraints 760 Here,fromc is theBBHs speed by of factors light, E ofand 1.1p areto the1.8 energy and 761 momentum of the GWs, and A↵ and ↵ are phenomenological 762 parameters. We consider ↵ values from 0 to 4 in steps of 0.5. 100 Two different ways of modifying PN 763 However, we exclude ↵ = 2, where the speed of the GWs is

764 | modifiedphasing in a frequency-independent terms: manner, and therefore i ˆ ' 12

765

| gives no observable dephasing. Thus, in all cases except 1 766 ↵ = 10 for 0,Enforce we are considering continuity a Lorentz-violating for post-inspiral dispersion GW150914 767 relation. The group velocity associated with this dispersion ↵ 2 2 GW151226 768 relation is(usedvg/c = for(dE /IMRPhenomPv2)dp)/c = 1 + (↵ 1)A↵E /2 + O(A↵). 1/(↵ 2) GW170104 769 The associated length scale is A B hc A↵ , where h 2 10 GW170608 | | 770 is Planck’s constant. A gives the scale of modifications to GW170814 Taper modification at high 771 the Newtonian potential (the Yukawa potential for ↵ = 0) Combined frequencies (used for Combined (SEOBNRv4) 772 associated with this dispersion relation. 3 773 10 While Eq.SEOBNRv4_ROM)(2) is a purely phenomenological model, it en- ) ) (l (l 774 compasses a variety of more fundamental predictions (at least

1PN 0PN .5PN 1PN .5PN 2PN 3PN .5PN 775 0 1 .5PN 3PN 3 to leading order) [94, 100]. In particular, A0 > 0 corre- 2 Constraints obtained from the two GW170817 alone gives 2 orders of magnitude 776 sponds to a massive graviton, i.e., the same dispersion as 777 for amethods massive particle in general in vacuo agreement [102], with a graviton mass FIG. 4.better 90% upper constraint bounds on theon absolute-1PN coeff; magnitude other of the GR- 1/2 2 13 778 given by mg = A0 /c . Furthermore, ↵ values of 2.5, 3, violating parameters 'ˆ n, from 1PN through 3.5PN in the inspiral 15 constraints comparable 779 and 4 correspond to the leading predictions of multi-fractal phase. At each PN order, we show results obtained from each of 780 spacetime [103]; doubly special relativity [104]; and Horava-ˇ the events listed in Table I that cross the SNR threshold in the inspi- 781 Lifshitz [105] and extra dimensional [106] theories, respec- ral regime, analyzed with IMRPhenomPv2. Bounds obtained from combining posteriors of events detected with a significance that ex- 782 tively. The standard model extension also gives a leading contri- 1 783 ↵ = ceeds a threshold of FAR < (1000 yr) in both modelled searches bution with 4[107], only considering the non-birefringent are shown for both analyses, using IMRPhenomPv2 (filled diamonds) 784 terms; our analysis does not allow for birefringence. and SEOBNRv4 (empty diamonds). 785 In order to obtain a waveform model with which to con- 786 strain these propagation e↵ects, we start by assuming that 787 the waveform extracted in the binary’s local wave zone (i.e., 788 near to the binary compared to the distance from the binary 737 constraint obtained on a positive PN coecient from a single 789 to Earth, but far from the binary compared to its own size) is 738 binary black hole event, as shown in Fig. 4. However, the con- 14 790 well-described by a waveform in GR. Since we are able to 739 straint at this order is about five times worse than that obtained 791 bound these propagation e↵ects to be very small, we can work 740 from the binary neutron star event GW170817 alone [8]. The 792 to linear order in A↵ when computing the e↵ects of this disper- 741 1PN bound is two orders of magnitude better for GW170817 15 793 sion on the frequency-domain GW phasing, thus obtaining a 742 than the best bound obtained from GW170608. For all other 794 correction [100] that is added to ( f ) in Eq. (1): 743 PN orders, GW170608 also provides the best bounds, which at 744 high PN orders are of the same order of magnitude as the ones ↵ 1 ⇡DL ↵ 2 f 745 from GW170817. Our results can be compared statistically to , ↵ , 1 ↵ A,e↵ 746 those obtained by performing the same tests on simulated GR 1 c ↵( f ) = sign(A↵) 8 ! . (3) > det 747 and non-GR waveforms given in [93]. The results presented > ⇡DL ⇡G f > ln M , ↵ = 1 748 here are consistent with those of GR waveforms injected into <> c3 > A,e↵ ! 749 realistic detector data. The combined bounds are the tightest > > 750 obtained so far, improving on the bounds obtained in [5] by :> 751 factors between 1.1 and 1.8. 12 For a source with an electromagnetic counterpart, A2 can be constrained by comparison with the arrival time of the photons, as was done with GW170817/GRB170817A [101]. 13 Thus, the Yukawa screening length is 0 = h/(mgc). 752 VII. PARAMETERIZED TESTS OF GRAVITATIONAL 14 This is likely to be a good assumption for ↵ < 2, where we constrain A to 753 WAVE PROPAGATION be much larger than the size of the binary. For ↵ > 2, where we constrain A to be much smaller than the size of the binary, one has to posit a screening mechanism in order to be able to assume that the waveform in the binary’s 754 We now place constraints on a phenomenological modifi- local wave zone is well-described by GR, as well as for this model to evade 755 cation of the GW dispersion relation, i.e., on a possible fre- Solar System constraints. 756 quency dependence of the speed of GWs. This modification, 15 The dimensionless parameter controlling the size of the linear correction ↵ 2 18 757 introduced in [100] and first applied to LIGO data in [6], is is A↵ f , which is . 10 at the 90% credible level for the events we consider and frequencies up to 1 kHz. 758 obtained by adding a power-law term in the momentum to the PARAMETERIZED TEST OF GW GENERATION: COMBINED CONSTRAINTS

leading tail and spin contributions 9

0.01 0.2 0.8 6 2 IMRPhenomPv2 0.005 0.1 0.4 3 1 SEOBNRv4 i ˆ p 0 0 0 0 0 -0.005 -0.1 -0.4 -3 -1 -0.01 -0.2 -0.8 -6 -2 ' 2 '0 '1 '2 '3 '4 '5l '6 '6l '7 2 3 ↵2 ↵3 ↵4

1 FIG. 3. Combined posteriors for parametrized violations of GR, obtained from all events in Table I with a significance of FAR < (1000 yr) in both modeled searches. The combined posteriorsinspiral on 'i in the inspiral regime are obtained from the events which inmerger- addition exceed the SNR threshold in the inspiral regime (GW150914, GW151226, GW170104, GW170608, and GW170814), analyzed with IMRPhenomPv2 (grey shaded region) and SEOBNRv4 (black outline). The combined posteriors on the intermediate and merger-ringdownringdown parameters i and ↵i are obtained from events which exceed the SNR threshold in the post-inspiral regime (GW150914, GW170104, GW170608, GW170809, GW170814, and GW170823), analyzed with IMRPhenomPv2. intermediate

651 flux are negligible, the contribution of higher-order terms can 694 In addition, allowing for a larger parameter space by varying 652 be significant in the binary black-hole signals that we study 695 multiple coecients simultaneously would not improve our 16 653 here. This prohibits an exact interpretation of the 1PN term 696 eciency in identifying violations of GR, as it would yield less 654 as the strength of dipolar radiation. Hence, this analysis only 697 informative posteriors. A specific alternative theory of gravity 655 serves as a test of the presence of an e↵ective 1PN term in 698 would likely yield correlated deviations in many parameters, 656 the inspiral phasing, which is absent in GR. 699 including modifications that we have not considered here. This 657 To measure the above GR violations in the post-Newtonian 700 would be the target of an exact comparison of an alternative 658 inspiral, we employ two waveform models: (i) the analytical 701 theory with GR, which would only be possible if a complete, 659 frequency-domain model IMRPhenomPv2 which also provided 702 accurate description of the inspiral-merger-ringdown signal in 660 the natural parametrization for our tests and (ii) SEOBNRv4, 703 that theory was available. 661 which we use in the form of SEOBNRv4 ROM, a frequency- 704 We use priors uniform on pˆi and symmetric around zero. 662 domain, reduced-order-model of the SEOBNRv4 model. The 705 Figure 3 shows the combined posteriors of pˆi (marginal- 663 inspiral part of SEOBNRv4 is based on a numerical evolution 706 ized over all other parameters) estimated from the combi- 664 of the aligned-spin e↵ective-one-body dynamics of the binary, 707 nation of all the events that cross the significance threshold 1 665 while its post-inspiral evolution is calibrated against NR simu- 708 of FAR < (1000 yr) in both modeled searches; see Table I. 666 lations. Despite its non-analytical nature, SEOBNRv4 ROM 709 Events with SNR< 6 in the inspiral regime (parameters 'ˆi) or 667 can also be used to test the parametrized modifications of 710 in the post-inspiral regime (ˆi and ↵ˆ i for the intermediate and 668 the early inspiral defined above. Using the method presented 711 merger-ringdown parameters respectively) are not included in 669 in [8], we add deviations to the waveform phase correspond- 712 the results, since the data from those instances failed to pro- 670 ing to a given 'ˆi at low frequencies and then taper the cor- 713 vide useful constraints (see Sec. III for more details). This 671 rections to zero at a frequency consistent with the transition 714 SNR threshold, however, is not equally e↵ective in ensuring 672 frequency between early-inspiral and intermediate phases used 715 informative results for all cases; see Sec. 3 in the Appendix 673 by IMRPhenomPv2. The same procedure cannot be applied to 716 for a detailed discussion. In all cases considered, the posteri- 674 the later stages of the waveform, thus the analysis performed 717 ors are consistent with pˆi = 0 within statistical fluctuations. 675 with SEOBNRv4 is restricted to the post-Newtonian inspiral, 718 Bounds on the inspiral coecients obtained with the two dif- 676 cf. Fig. 3. 719 ferent waveform models are found to be in good agreement 677 The analytical descriptions of the intermediate and merger- 720 with each other. Finally, we note that the event-combining 678 ringdown stages in the IMRPhenomPv2 model allow for a 721 analyses on pˆi incorporate the highly non-trivial assumption 679 straightforward way of parametrizing deviations from GR, de- 722 that these parametrized violations are constant across all events 680 noted by ˆ , ˆ and ↵ˆ , ↵ˆ , ↵ˆ respectively, follow- 723 considered. While this is valid for our null hypothesis (GR), { 2 3} { 2 3 4} 681 ing [93]. Here the parameters ˆi correspond to deviations 724 in an active search for signs of modified gravity this assump- 682 from the NR-calibrated phenomenological coecients i of 725 tion should be dropped. Posterior distributions of pˆi for the 683 the intermediate stage, while the parameters ↵ˆ i refer to modi- 726 individual-event analysis, also showing full consistency with 684 fications of the merger-ringdown coecients ↵i obtained from 727 GR, are provided in Sec. 3 of the Appendix. 685 a combination of phenomenological fits and analytical black- 728 Figure 4 shows the 90% upper bounds on 'ˆ for all the | i| 686 hole perturbation theory calculations [19]. 729 individual events which cross the SNR threshold (SNR > 6) in 687 Using LALInference, we calculate posterior distributions of 730 the inspiral regime (the most massive of which is GW150914). 688 the parameters characterizing the waveform (including those 731 The bounds from the combined posteriors are also shown; 689 that describe the binary in GR). Our parametrization recovers 732 these include the events which exceed both the SNR thresh- 690 GR at pˆi = 0, so consistency with GR is verified if the poste- 733 old in the inspiral regime as well as the significance threshold, 691 riors of pˆi have support at zero. We perform the analyses by 734 namely GW150914, GW151226, GW170104, GW170608, and 692 varying one pˆi at a time; as shown in Ref. [99], this is fully 735 GW170814. The bound from the likely lightest mass binary 693 robust to detecting deviations present in multiple PN-orders. 736 black hole event GW170608 at 1.5PN is currently the strongest PARAMETERIZED TEST OF GW PROPAGATION: FORMULATION10

2 2 2 759 dispersion relationPhenomenologicalE = p modificationc of GWs to GW in GR, dispersion giving relation: 2 2 2 ↵ ↵ E = p c + A↵ p c . (2) α = 0, A > 0: Massive graviton 760 Here, c is theα = 2.5: speed Multifractal of light, spacetimeE and p are the energy and 761 momentum ofα = the3: Doubly GWs, special and relativityA↵ and ↵ are phenomenological α = 4: Hořava-Lifshitz and extra dimensions 762 parameters. We consider ↵ values from 0 to 4 in steps of 0.5. (all but massive graviton also predict additional terms 763 However, wewith exclude higher powers↵ = of2, p) where the speed of the GWs is 764 modified in a frequency-independent manner, and therefore Assume that signal close to12 source is given by GR, all 765 gives no observable dephasing. Thus, in all cases except modifications come from propagation 766 for ↵ = 0, we are considering a Lorentz-violating dispersion Leads to a dephasing of the entire GW signal, 767 relation. The group velocity associated with this dispersion proportional to Aα and (roughly) to the distance↵ 2travelled 2 768 relation is vg/c = (dE/dp)/c = 1 + (↵ 1)A↵E /2 + O(A↵). Introduced in Mirshekari et al. PRD (2012)1/(↵ 2) 17 769 The associated length scale is B hc A↵ , where h A | | 770 is Planck’s constant. A gives the scale of modifications to 771 the Newtonian potential (the Yukawa potential for ↵ = 0) 772 associated with this dispersion relation. 773 While Eq. (2) is a purely phenomenological model, it en- 774 compasses a variety of more fundamental predictions (at least 775 to leading order) [94, 100]. In particular, A0 > 0 corre- 776 sponds to a massive graviton, i.e., the same dispersion as 777 for a massive particle in vacuo [102], with a graviton mass FIG. 4. 90% upper bounds on the absolute magnitude of the GR- 1/2 2 13 778 given by mg = A0 /c . Furthermore, ↵ values of 2.5, 3, violating parameters 'ˆ n, from 1PN through 3.5PN in the inspiral 779 and 4 correspond to the leading predictions of multi-fractal phase. At each PN order, we show results obtained from each of 780 spacetime [103]; doubly special relativity [104]; and Horava-ˇ the events listed in Table I that cross the SNR threshold in the inspi- 781 Lifshitz [105] and extra dimensional [106] theories, respec- ral regime, analyzed with IMRPhenomPv2. Bounds obtained from combining posteriors of events detected with a significance that ex- 782 tively. The standard model extension also gives a leading contri- 1 783 ↵ = ceeds a threshold of FAR < (1000 yr) in both modelled searches bution with 4[107], only considering the non-birefringent are shown for both analyses, using IMRPhenomPv2 (filled diamonds) 784 terms; our analysis does not allow for birefringence. and SEOBNRv4 (empty diamonds). 785 In order to obtain a waveform model with which to con- 786 strain these propagation e↵ects, we start by assuming that 787 the waveform extracted in the binary’s local wave zone (i.e., 788 near to the binary compared to the distance from the binary 737 constraint obtained on a positive PN coecient from a single 789 to Earth, but far from the binary compared to its own size) is 738 binary black hole event, as shown in Fig. 4. However, the con- 14 790 well-described by a waveform in GR. Since we are able to 739 straint at this order is about five times worse than that obtained 791 bound these propagation e↵ects to be very small, we can work 740 from the binary neutron star event GW170817 alone [8]. The 792 to linear order in A↵ when computing the e↵ects of this disper- 741 1PN bound is two orders of magnitude better for GW170817 15 793 sion on the frequency-domain GW phasing, thus obtaining a 742 than the best bound obtained from GW170608. For all other 794 correction [100] that is added to ( f ) in Eq. (1): 743 PN orders, GW170608 also provides the best bounds, which at 744 high PN orders are of the same order of magnitude as the ones ↵ 1 ⇡DL ↵ 2 f 745 from GW170817. Our results can be compared statistically to , ↵ , 1 ↵ A,e↵ 746 those obtained by performing the same tests on simulated GR 1 c ↵( f ) = sign(A↵) 8 ! . (3) > det 747 and non-GR waveforms given in [93]. The results presented > ⇡DL ⇡G f > ln M , ↵ = 1 748 here are consistent with those of GR waveforms injected into <> c3 > A,e↵ ! 749 realistic detector data. The combined bounds are the tightest > > 750 obtained so far, improving on the bounds obtained in [5] by :> 751 factors between 1.1 and 1.8. 12 For a source with an electromagnetic counterpart, A2 can be constrained by comparison with the arrival time of the photons, as was done with GW170817/GRB170817A [101]. 13 Thus, the Yukawa screening length is 0 = h/(mgc). 752 VII. PARAMETERIZED TESTS OF GRAVITATIONAL 14 This is likely to be a good assumption for ↵ < 2, where we constrain A to 753 WAVE PROPAGATION be much larger than the size of the binary. For ↵ > 2, where we constrain A to be much smaller than the size of the binary, one has to posit a screening mechanism in order to be able to assume that the waveform in the binary’s 754 We now place constraints on a phenomenological modifi- local wave zone is well-described by GR, as well as for this model to evade 755 cation of the GW dispersion relation, i.e., on a possible fre- Solar System constraints. 756 quency dependence of the speed of GWs. This modification, 15 The dimensionless parameter controlling the size of the linear correction ↵ 2 18 757 introduced in [100] and first applied to LIGO data in [6], is is A↵ f , which is . 10 at the 90% credible level for the events we consider and frequencies up to 1 kHz. 758 obtained by adding a power-law term in the momentum to the 11

det 795 Here, D is the binary’s luminosity distance, is the bi- L M 796 nary’s detector-frame (i.e., redshifted) chirp mass, and A,e↵ 797 is the e↵ective wavelength parameter used in the sampling, 798 defined as

1 ↵ 1/(↵ 2) (1 + z) DL , ↵ B . (4) A e D A " ↵ #

799 The parameter z is the binary’s redshift, and D↵ is a distance 800 parameter given by

1 ↵ z ↵ 2 (1 + z) (1 + z¯) D↵ = dz¯ , (5) H 3 0 Z0 ⌦m(1 + z¯) + ⌦⇤ 1 p1 801 where H0 = 67.90 km s Mpc is the Hubble constant, and 802 ⌦m = 0.3065 and ⌦⇤ = 0.6935 are the matter and dark energy 803 density parameters; these are the TT+lowP+lensing+ext values 16 804 from [108]. FIG. 5. 90% credible upper bounds on the absolute value of the modi- 805 The dephasing in Eq. (3) is obtained by treating the gravita- fied dispersion relation parameter A↵. We show results for positive 806 tional wave as a stream of particles (gravitons), which travel and negative values of A↵ separately. Specifically, we give the up- ↵ 2 2 807 at the particle velocity vp/c = pc/E = 1 A↵E /2 + O(A ). dated versions of the results from combining together GW150914, ↵ 808 There are suggestions to use the particle velocity when consid- GW151226, and GW170104 (first given in [6]), as well as the re- 809 ering doubly special relativity, though there are also sugges- sults from combining together all the events meeting our significance 810 tions toPARAMETERIZED use the group velocity vg in that case (see, e.g., [110] threshold TEST for combined OF results (see TableGWI). Picoelectronvolts (peV) provide a convenient scale, because 1 peV h 250 Hz, where 811 and references therein for both arguments). However, the group ' ⇥ 250 Hz is roughly around the most sensitive frequencies of the LIGO 812 velocity is appropriate for, e.g., multi-fractal spacetime theo- 11 and Virgo instruments. 813 ries (see,PROPAGATION: e.g., [111]). To convert the bounds presented here to RESULTS 814 the case where the particles travel at the group velocity, scale det A↵ < 0 A↵ > 0 795 Here, DL is the binary’s luminosity distance, is the bi- 2 1.0 M 815 the A↵ bounds for ↵ , 1 by factors of 1/(1 ↵). The group 796 nary’s detector-frame (i.e., redshifted) chirp mass, and A,e↵ 816 velocity calculation gives an unobservable constant phase shift 0.4 797 is the e↵ective wavelength parameter used in the sampling, 19 817 for ↵ =101. 798 defined as 818 We consider the cases of positive and negative A↵ separately, 1 0.5 0.2 1 ↵ 1/(↵ 2) 819 and obtain the results shown in Table IV and Fig. 5 when ] ↵ + ] (1 z) DL ↵

820 2 A,e↵ B A . (4) applying this analysis to the GW events under consideration. 2 D↵ 821 " # While we sample with a flat prior in log A,e↵, our bounds are peV 0.0 0 0.0 822 A 19 given[peV using priors flat in ↵ for all results except for the mass of 799 The parameter z is the binary’s redshift, and D↵ is a distance 20 823 the graviton,10 where we use a prior flat in the graviton mass. We | 800 [10 parameter given by ↵ 824 ↵ also showA the results from combining together all the signals A | 0.2 1 ↵ z ↵ 2 825 that satisfy our selection criterion. We are able to combine 1 0.5 (1 + z) (1 + z¯) D↵ = dz¯ , (5)826 together the results from di↵erent signals with no ambiguity, 3 H0 0 ⌦m(1 + z¯) + ⌦⇤ Z 827 since the known distance dependence is accounted for in the 0.4 GW150914 + GW151226 + GW170104 828 waveforms. O1 and O2 combined results 1 p1 2 1.0 801 where H0 = 67.90 km s Mpc is the Hubble constant, and 21 829 Figure10 6 displays the full A posteriors obtained by combin- 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0 1 2 3 ↵ 4 0 1 2 3 4 802 ⌦ = 0.3065 and ⌦⇤ = 0.6935 are the matter and dark energy ↵ m 830 ing all selected events↵ (using IMRPhenomPv2 waveforms).↵ To 803 density parameters; these are the TT+lowP+lensing+ext values 831 A 16 obtain theImprovements full ↵ posteriors, of up we to combine a factor together of 2.4 on the previous positive results (due to including a number of more distant systems). 804 from [108]. 832 and negative A results for individual events by weighting by FIG. 6. Violin plots of the full posteriors on the modified dispersion FIG. 5. 90% credible↵ upper bounds on the absolute value of the modi- -23 805 The dephasing in Eq. (3) is obtained by treating the gravita- relation parameter A↵ calculated from the combined events, with the 833 theirfied Bayesian dispersionMassive evidences; relation graviton parameter we improved then combineA↵. by We a show factor the posteriors results of 1.5: for ≤ from positive 5.0 x 10 eV (90% credible level) 806 tional wave as a stream of particles (gravitons), which travel -23 90% credible interval around the median indicated. 834 and negative values of A↵ separately. Specifically, we≤ give the up- ↵ 2 2 individualsolar events. system We give bound the analogous(Yukawa potential): plots for the 6.76 individ- x 10 eV (90% confidence level) [Bernus et al., arXiv (2019)] 807 at the particle velocity vp/c = pc/E = 1 A↵E /2 + O(A ). dated versions of the results from combining together GW150914, ↵835 ual events in Sec. 4 of the Appendix. The combined positive 808 There are suggestions to use the particle velocity when consid- GW151226,Can andcombine GW170104 results (first with given no assumptions in [6]), as well beyond as the those re- used to construct test 18 836 and negative A↵ posteriors are also used to compute the GR 809 ering doubly special relativity, though there are also sugges- sults from combining together all the events meeting our significance838 A↵ < 0, where A↵ = 0 is the GR value. Thus, large or small 837 quantiles given in Table IV, which give the probability to have 810 tions to use the group velocity vg in that case (see, e.g., [110] threshold for combined results (see Table I). Picoelectronvolts (peV)839 values of the GR quantile indicate that the distribution is not provide a convenient scale, because 1 peV h 250 Hz, where840 peaked close to the GR value. For a GR signal, the GR quan- 811 and references therein for both arguments). However, the group ' ⇥ 250 Hz is roughly around the most sensitive frequencies of the LIGO 812 velocity is appropriate for, e.g., multi-fractal spacetime theo- 841 tile will be distributed uniformly in [0, 1] for di↵erent noise and Virgo instruments. 842 realizations. The GR quantiles we find are consistent with 813 ries (see, e.g., [111]). To convert the bounds presented here to 16 We use these values for consistency with the results presented in [14]. 814 the case where the particles travel at the group velocity, scale If we instead use the more recent results from [109], specifically the 843 such a uniform distribution. In particular, the (two-tailed) meta 815 the A↵ bounds for ↵ , 1 by factors of 1/(1 ↵). The group TT,TE,EE+lowE+lensing+BAO values used for comparison in [14], then 844 p-value for all events and ↵ values obtained using Fisher’s 816 velocity calculation gives an unobservable constant phase shift there are very minor changes to the results presented in this section. For 845 method [74] (as in Sec. VA) is 0.9995. instance, the upper bounds in Table IV change by at most 0.1%. 817 for ↵ = 1. ⇠ 846 We find that the combined bounds overall improve on those 818 We consider the cases of positive and negative A↵ separately, 819 and obtain the results shown in Table IV and Fig. 5 when 820 applying this analysis to the GW events under consideration. 821 While we sample with a flat prior in log A,e↵, our bounds are 822 given using priors flat in A↵ for all results except for the mass of 823 the graviton, where we use a prior flat in the graviton mass. We 824 also show the results from combining together all the signals 825 that satisfy our selection criterion. We are able to combine 826 together the results from di↵erent signals with no ambiguity, 827 since the known distance dependence is accounted for in the 828 waveforms. 829 Figure 6 displays the full A↵ posteriors obtained by combin- 830 ing all selected events (using IMRPhenomPv2 waveforms). To 831 obtain the full A↵ posteriors, we combine together the positive 832 and negative A↵ results for individual events by weighting by FIG. 6. Violin plots of the full posteriors on the modified dispersion relation parameter A↵ calculated from the combined events, with the 833 their Bayesian evidences; we then combine the posteriors from 90% credible interval around the median indicated. 834 individual events. We give the analogous plots for the individ- 835 ual events in Sec. 4 of the Appendix. The combined positive 836 and negative A↵ posteriors are also used to compute the GR 838 A↵ < 0, where A↵ = 0 is the GR value. Thus, large or small 837 quantiles given in Table IV, which give the probability to have 839 values of the GR quantile indicate that the distribution is not 840 peaked close to the GR value. For a GR signal, the GR quan- 841 tile will be distributed uniformly in [0, 1] for di↵erent noise 842 realizations. The GR quantiles we find are consistent with 16 We use these values for consistency with the results presented in [14]. If we instead use the more recent results from [109], specifically the 843 such a uniform distribution. In particular, the (two-tailed) meta TT,TE,EE+lowE+lensing+BAO values used for comparison in [14], then 844 p-value for all events and ↵ values obtained using Fisher’s there are very minor changes to the results presented in this section. For 845 method [74] (as in Sec. VA) is 0.9995. instance, the upper bounds in Table IV change by at most 0.1%. ⇠ 846 We find that the combined bounds overall improve on those The Confrontation between General Relativity and Experiment 81

Three modes (A+, A , and AS) are transverse to the direction of propagation, with two repre- senting quadrupolar deformations⇥ and one representing a monopolar transverse “breathing” de- formation. Three modes are longitudinal, with one (AL) an axially symmetric stretching mode in the propagation direction, and one quadrupolar mode in each of the two orthogonal planes POLARIZATION CONSTRAINTScontaining the propagation direction (AV1 and AV2). Figure 10 shows the displacements induced on a ring of freely falling test particles by each of these modes. General relativity predicts only the first two transverse quadrupolar modes (a) and (b) independently of the source; these corre- spond to the waveforms h+ and h discussed earlier (note the cos 2 and sin 2 dependences of the displacements). ⇥

Gravitational−Wave Polarization y y With three detectors, can distinguish between

purely scalar, purely vector, and purely tensor x x polarizations, from the different beam patterns. (a) (b) y y GW170817 gives by far the best constraints, due 20 to the EM localization (Bayes factors > 10 ), but x z can also obtain constraints from sufficiently loud (c) (d) and well-localized BBHs: x y

BF: Tensor/Scalar BF: Tensor/Vector z z GW170814 220 ± 27 30 ± 4 (e) (f) Figure 10: The six polarization modes for gravitational waves permitted in any metric theory of gravity. Shown is the displacementGW polarizations that each mode induces in on aa ring general of test particles. theory The wave propagates in GW170818 407 ± 100 12 ± 3 the +z direction. There is no displacement out of the plane of the picture. In (a), (b), and (c), the wave propagates out of the plane; in (d),from (e), and Will (f), the LRR wave propagates 2014 in the plane. In GR, only (a) and Method described in Isi and Weinstein, arXiv (2017) (b) are present; in massless scalar–tensor gravity, (c) may also be present. 19 Massless scalar–tensor gravitational waves can in addition contain the transverse breathing mode (c). This can be obtained from the physical waveform h↵, which is related to h˜↵ and ' to

Living Reviews in Relativity http://www.livingreviews.org/lrr-2014-4 CONCLUSIONS

The ten BBHs observed by LIGO and Virgo in their first two observing runs allow us to make tests of GR in extreme—but quite clean—environments

Four tests carried out on the data:

Residuals, IMR consistency, param. generation, param. propagation

No evidence for departures from GR—combined results improved by factors from 1.1 to 2.4

Also checked polarization constraints—much less constraining than GW170817.

Results publicly available for further exploration: https://dcc.ligo.org/LIGO-P1900087/public 20 OUTLOOK

Future prospects for such tests are very good:

We expect many more (and louder) signals with improved sensitivity in O3 and beyond.

Many new tests are being developed and implemented.

GR waveform models are being improved in accuracy and physics included.

There is preliminary work on creating waveforms (and waveform models) beyond GR.

Such models will be necessary to make the best use of the data for testing GR. 21 THANK YOU!

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