Numerical relativity simulation of GW150914 in Einstein dilaton Gauss-Bonnet gravity
NNumerical relativity simulation of GW150914 in Einstein dilaton Gauss-Bonnetgravity
Maria Okounkova Center for Computational Astrophysics, Flatiron Institute, 162 5th Ave, New York, NY 10010 ∗ (Dated: January 13, 2020)A present challenge in testing general relativity (GR) with binary black hole gravitational wavedetections is the inability to perform model-dependent tests due to the lack of merger waveformsin beyond-GR theories. In this study, we produce the first numerical relativity binary black holegravitational waveform in Einstein dilaton Gauss-Bonnet (EDGB) gravity, a higher-curvature theoryof gravity with motivations in string theory. We evolve a binary black hole system in order-reducedEDGB gravity, with parameters consistent with GW150914. We focus on the merger portion of thewaveform, due to the presence of secular growth in the inspiral phase. We compute mismatches withthe corresponding general relativity merger waveform, finding that from a post-inspiral-only analysis,we can constrain the EDGB lengthscale to be √ α GB (cid:46) km. I. INTRODUCTION
Though Einstein’s theory of general relativity (GR) haspassed all precision tests to date, at some lengthscale, itmust break down and be reconciled with quantum mechan-ics in a beyond-GR theory of gravity. Binary black hole(BBH) mergers probe the strong-field, non-linear regimeof gravity, and thus gravitational wave signals from thesesystems could contain signatures of a beyond-GR the-ory. While LIGO presently performs model-independent and parametrized tests of general relativity [1, 2], one im-portant additional avenue of looking for deviations fromgeneral relativity is to perform model-dependent tests.Such model-dependent tests require access to numeri-cal waveforms in beyond-GR theories of gravity throughmerger, the lack of which is currently a severe limitationon constraining beyond-GR physics [3].We produce the first numerical relativity gravitationalwaveforms in Einstein dilaton Gauss-Bonnet (EDGB)gravity, an effective field theory that modifies the Einstein-Hilbert action of GR through the inclusion of a scalar fieldcoupled to terms quadratic in curvature. These terms aremeant to encompass underlying quantum gravity effects,in particular motivated by string theory [4–7], and thecoupling to the scalar field is governed by an EDGBlengthscale parameter √ α GB . The well-posedness of theinitial value problem in full EDGB gravity is unknown [8–11]. We thus work in an order-reduction scheme , in whichwe perturb the EDGB scalar field and spacetime metricabout a GR background.Previously, Witek et al. [12] evolved the leading-orderEDGB scalar field on a BBH background, predicting abound of √ α GB (cid:46) . km on the EDGB lengthscale, aconstraint seven orders of magnitude tighter than obser-vational results from solar-system tests. In this study, weevolve the leading-order EDGB correction to the space-time metric on a BBH background, thus obtaining the ∗ mokounkova@flatironinstitute.org leading-order EDGB modification to the merger gravita-tional waveform . We compute mismatches between theGR and EDGB-corrected waveforms, aiming to similarlybound the EDGB lengthscale.We focus on an astrophysically-relevant BBH systemwith spin and mass ratio consistent with GW150914, theloudest LIGO detection to date [13–15], for which signif-icant model-independent and parametrized tests of GRhave been performed [1–3, 16, 17]. This extends ourresults in [18], where we simulated the same system in dy-namical Chern-Simons gravity (dCS), another quadraticbeyond-GR theory with motivations in string theory andloop quantum gravity [19–22]. II. SETUP
We set G = c = 1 throughout. Quantities are given interms of units of M , the sum of the Christodolou masses ofthe background black holes at a given reference time [23].Latin letters in the beginning of the alphabet { a, b, c, d . . . } denote 4-dimensional spacetime indices, and g ab refers tothe spacetime metric with covariant derivative ∇ c . A. Equations of motion
The overall form of the EDGB action that we will use inthis paper, chosen to be consistent with Witek et al. [12],is S ≡ (cid:90) m pl d x √− g (cid:20) R −
12 ( ∂ϑ ) + 2 α GB f ( ϑ ) R GB (cid:21) , (1)where the first term is the Einstein-Hilbert action of GR(where R is the 4-dimensional Ricci scalar), ϑ is the EDGBscalar field, and α GB is the EDGB coupling parameterwith dimensions of length squared. We will work with √ α GB , which has dimensions of length, throughout this a r X i v : . [ g r- q c ] J a n paper. Finally, R GB is the EDGB scalar, of the form R GB = R abcd R abcd − R ab R ab + R . (2)It is unknown whether EDGB has a well-posed initialvalue problem [8–11]. However, as we have done in [18, 24–26], we perturb the spacetime metric and EDGB scalarfield about an arbitrary GR background as g ab = g (0) ab + ∞ (cid:88) n =1 ε n g ( n ) ab , (3) ϑ = ∞ (cid:88) n =0 ε n ϑ ( n ) , (4)where ε is an order-counting parameter that counts pow-ers of α GB , and superscript (0) corresponds to the GRsolution, which we refer to as the background .At each order, the equations of motion are well-posed.Moreover, the EDGB coupling parameter α GB scales outat each order, and thus we only need to perform one BBHsimulation for each set of GR background parameters.Zeroth order corresponds to pure general relativity. Theequation of motion for the zeroth order scalar field, ϑ (0) ,corresponds to a scalar field minimally coupled to vacuumGR, and thus ϑ (0) should decay to zero in BH spacetimesby the no-hair theorem.The leading-order EDGB scalar field appears at first-order as ϑ (1) , sourced by the curvature of the GR back-ground, with equation of motion (cf. [12] for a full deriva-tion), (cid:3) (0) ϑ (1) = − M R (0)GB , (5) R (0)GB ≡ R (0) abcd R (0) abcd − R (0) ab R (0) ab + R (0)2 , (6)where the superscript (0) refers to quantities computedfrom the GR background. Here, the leading-order correc-tion to f ( ϑ ) has been set to , in accordance with [12].Meanwhile, the leading EDGB deformation to the space-time metric comes in at second order, with the equationof motion (cf. [12]), G (0) ab [ g (2) ab ] = − M G (0) ab [ ϑ (1) ] + T ab [ ϑ (1) ] . (7)In the above equations, T ab [ ϑ (1) ] is the standard Klein-Gordon stress energy tensor associated with ϑ (1) , of theform T ab [ ϑ (1) ] = ∇ (0) a ϑ (1) ∇ (0) b ϑ (1) − g (0) ab ∇ (0) c ϑ (1) ∇ (0) c ϑ (1) , (8) Note that our definition of T ab [ ϑ (1) ] in Eq. (8) differs from Eq.15 in [12] by a factor of , and hence the T ab [ ϑ (1) ] term in Eq. (7)differs by a factor of . We have chosen this convention to be inline with the canonical form of the Klein-Gordon stress-energytensor. and G (0) ab [ ϑ (1) ] = 2 (cid:15) edfg g (0) c ( a g (0) b ) d ∇ (0) h (cid:20) ∗ R (0) chfg ∇ (0) e ϑ (1) (cid:21) , (9)where ∗ R abcd = (cid:15) abef R (0) efcd and (cid:15) abcd is the Levi-Citivapseudo-tensor, with (cid:15) abcd = − [ abcd ] / (cid:112) − g (0) , where [ abcd ] is the alternating symbol.Note that we work on a vacuum GR background, andthus terms vanish to give the simplified equations of mo-tion (cid:3) (0) ϑ (1) = − M R (0) abcd R (0) abcd (10) G (0) ab [ g (2) ab ] = − M (cid:15) edfg g (0) c ( a g (0) b ) d ∗ R (0) chfg ∇ (0) h ∇ (0) e ϑ (1) + T ab [ ϑ (1) ] . (11) To summarize: the order-reduction procedure is illus-trated in Fig. 1 of [18]. We will have a GR binary blackhole background. The curvature of this background willthen source the leading-order EDGB scalar field (Eq. (10)).This leading-order scalar field and the GR backgroundwill then source the leading-order EDGB correction tothe spacetime (Eq. (11)), which in turn will give us theleading-order EDGB correction to the gravitational wave-form.
B. Secular growth during inspiral
As we initially noted in [18], the perturbative order-reduction scheme outlined in Sec. II A gives rise to seculargrowth during the inspiral. In the order-reduction scheme,the rate of inspiral is governed by the GR background.However, in the full, non-linear EDGB theory, we expectthe black holes to have a faster rate of inspiral due toenergy loss to the scalar field [27]. Since we do not backre-act on the GR background in the order-reduction scheme,we do not capture this correction to the rate of inspiral,and hence our solution contains secular growth. This is afeature generically found in perturbative treatments [28],including in extreme mass-ratio inspirals [29].When we simulated an inspiraling binary black holesystem in order-reduced dCS [18], we indeed observedsecular growth during the inspiral. We performed a setof simulations where we ramped on the dCS source termsat various start times during the inspiral, for the sameset of background parameters. We found secular growthin the amplitude of the resulting dCS correction to thewaveform, with simulations with earlier start times havinglarger amplitudes.
However , this secular growth settled toa quadratic minimum for a start time before the portionof the inspiral-merger present in the LIGO band for aGW150914-like system. Thus, we were able to focuson this portion of the waveform in [18] without havingcontamination from secular effects.In this study, we apply the same procedure, where weramp on the EDGB source terms at a variety of start timesfor the same (long) GR binary black hole backgroundsimulation. We search for the start time at which thewaveform is no longer contaminated by secular effects,and present the resulting merger waveform.The inspiral in EDGB is more strongly modified fromGR, with the modifications to the inspiral occurring at -1PN order relative to GR due to the presence of dipolarradiation in the scalar field [27]. This is 3 PN ordershigher than the leading modification in the dCS case,where dipolar radiation is absent during inspiral. Thuswe expect the minimum of the secular growth to occurlater in the inspiral in EDGB than in dCS for the samephysical system.
C. Computational details
Eqs. (10) and (11) are precisely the equations that weco-evolve with the GR background. We use the SpectralEinstein Code [30], which uses pseudo-spectral methodsand thus guarantees exponential convergence in the fields.All of the technical details are given in [18, 24–26]. Thedomain decomposition is precisely that of the analogousdCS study [18].
III. EDGB MERGER WAVEFORMSA. Simulation parameters
While there is a distribution of mass and spin param-eters consistent with GW150914 [14, 31], we choose touse the parameters of SXS:BBH:0305, as given in theSimulating eXtreme Spacetimes (SXS) catalog [32]. Thissimulation was used in Fig. 1 of the GW150914 detectionpaper [13], as well a host of follow-up studies [33–35]. Weadditionally used precisely these parameters for our dCSBBH simulation [18]. The configuration has initial dimen-sionless spins χ A = 0 . z and χ B = − . z , alignedand anti-aligned with the orbital angular momentum. Thedominant GR spherical harmonic modes of the gravita-tional radiation for this system are ( l, m ) = (2 , ± . Thesystem has initial masses of . M and . M , lead-ing to a mass ratio of . . The initial eccentricity is ∼ × − . The remnant has final Christodolou mass . M and dimensionless spin . purely in the ˆ z direction. The GR background simulation completes 23orbits before merger. B. Regime of validity
The results that we present for the leading-order EDGBscalar and gravitational waveforms have the EDGB cou-pling parameter α GB scaled out. For the perturbativeorder reduction scheme to be valid, we require that g (2) ab (cid:46) Cg (0) ab , for some constant C < . This in turn becomes a constraint on α GB , of the form (cf. [24] for ananalogous derivation) √ α GB GM (cid:46) (cid:32) C (cid:107) g (0) ab (cid:107)(cid:107) g (2) ab (cid:107) (cid:33) / . (12)We choose C = 0 . , and evaluate Eq. (12) on each sliceof the numerical relativity simulation. We find the thestrongest constraint on the allowed value of √ α GB /GM comes at merger, when the spacetime is most highlyperturbed, with a value of √ α GB /GM ∼ . for thesimulation presented in this paper. C. EDGB scalar field waveforms
In Fig. 1, we show the results for the leading-orderEDGB scalar field, ϑ (1) . We decompose the scalar fieldinto spherical harmonics, and find that the dominantmodes are ( l, m = l ) , in accordance with [12, 27]. Wesee the presence of l = 1 dipolar radiation in the fieldduring inspiral, in accordance with [12, 27]. We see thatthe monopolar l = 0 mode is non-radiative during theinspiral, but that there is a burst of monopolar radiationat merger. This is in agreement with [12], and moreoveris similar to the results in dCS [24], where we found thatthe leading non-radiative mode (the dipole in the dCScase) exhibits a burst of radiation at merger. D. EDGB gravitational waveforms
As explained in Sec. II B, because of secular growthduring the inspiral, we focus on simulations with EDGBeffects ramped on close to merger, in order to mitigatethe amount of secular growth from the inspiral (we givemore details in Sec. III E). We thus present these mergerwaveforms in this section.From the leading-order EDGB metric deformation g (2) ab ,we can compute Ψ (2)4 , the leading-order modification tothe gravitational waveform, given by the Newman-Penrosescalar Ψ . Note that g (2) ab and hence Ψ (2)4 from the simu-lation are independent of the EDGB coupling parameter.In order to produce a full, second-order-accurate EDGBgravitational waveform, we must add Ψ (2)4 to the back-ground GR waveform Ψ (0)4 as Ψ = Ψ (0)4 + ( √ α GB /GM ) Ψ (2)4 + O (( √ α GB /GM ) ) , (13)for a given choice for the EDGB coupling parameter √ α GB /GM . We require that √ α GB /GM lies within theregime of validity for the perturbative scheme as given inSec. III BIn Fig. 2, we show this total waveform for a variety ofvalues of √ α GB /GM . We see that the EDGB-correctedwaveform has an amplitude shift relative to GR, as well − × − r Ψ (0)4 (2 , − × R e M o d e a m p li t ud e ( √ α GB /GM ) − rϑ (1) (0, 0) − × − ( √ α GB /GM ) − rϑ (1) (1, 1) − − − −
50 0 50 100( t − t peak ) /M − × − ( √ α GB /GM ) − rϑ (1) (2, 2) FIG. 1. Dominant modes of the leading-order EDGB scalarfield ϑ (1) , decomposed into spherical harmonics ( l, m ) , as afunction of time relative to the peak time of the GR gravita-tional waveform. The top panel corresponds to the dominant (2 , mode of the GR gravitational radiation for comparison.The bottom three panels correspond to the dominant modes of ϑ (1) , which are ( l, m = l ) . We see the presence of l = 1 dipolarradiation during the inspiral. While the l = 0 monopole isnon-radiative during the inspiral, we see a burst of monopolarradiation at merger. Compare with Fig. 4 of [12] and the dCScase in Fig. 1 of [24]. Note that the ϑ (1) waveforms have theEDGB coupling √ α GB /GM scaled out, and thus an appropri-ate value (cf. Sec. III B) of this coupling parameter must bere-introduced for the results to be physically meaningful. as a phase shift, consistent with the notion that EDGBshould have a faster inspiral due to energy loss to thescalar field [27]. − − −
20 0 20 40 60( t − t peak ) /M − . − . − . . . . . R e Ψ ( ) ( , ) + ( √ α G B / G M ) Ψ ( ) ( , ) ( √ α GB /GM ) FIG. 2. EDGB-corrected merger gravitational waveforms,as computed from Eq. (13), for a variety of values of theEDGB coupling parameter √ α GB /GM . The dashed blackline, with √ α GB /GM = 0 , corresponds to the GR waveform.The value √ α GB /GM = 0 . corresponds to the maximalallowed value in order for the perturbative scheme to be valid(cf. Sec. III B). We see that the EDGB-corrected waveformhas both an amplitude and phase shift relative to GR. E. Secular growth
As discussed in Sec. II B, the perturbative scheme leadsto secular growth in the inspiral waveform. In Fig. 3, weshow the leading-order EDGB correction to the gravita-tional waveform for a variety of simulation lengths (withthe same background GR simulation). We ramp on theEDGB source terms at different start times in order toproduce different inspiral lengths, as discussed in Sec. II B.We see that the longest simulations have the largest am-plitude at merger, consistent with secular growth. InFig. 4, we take a more quantitative look, plotting thepeak amplitude of the waveform as a function of inspirallength. In the dCS case (cf. Fig. 7 of [18]), we saw thatfor the closest start time to merger, the secular growth at-tained a quadratic minimum. In other words, the mergerwaveform we presented was not contaminated by seculareffects.In Fig. 4, we see a similar quadratic minimum for theEDGB correction to the waveform, although this occursat a shorter inspiral length (later start time) than in dCS.This higher level of secular growth in EDGB than in dCSis consistent with the theoretical predictions of Sec. II B,as the EDGB inspiral is more heavily modified than indCS due to the presence of dipolar radiation [27].
IV. CONSTRAINTS ON √ α GB FROM EDGBMERGER WAVEFORMS
As shown in Sec. III D, we have access to the leading-order EDGB merger waveform for a GW150914-like sys- − −
50 0 50 100( t − t peak ) /M − R e ( √ α G B / G M ) − r Ψ ( ) ( , ) Inspiral Length [M]266228204 179157
FIG. 3. Secular growth in leading-order EDGB gravitationalwaveforms as function of inspiral length of the waveform. Eachcolored curve corresponds to a simulation with a different starttime for the EDGB fields (as discussed in Sec. II B), with thesame GR background simulation for each. We label each curveby the time difference between the peak of the waveform andthe start time of ramping on the EDGB field (minus the ramptime). We see that simulations with earlier EDGB start timeshave higher amplitudes at merger, having had more time toaccumulate secular growth. tem. What sort of physical constraints on EDGB can weextract from the merger phase?
A. Merger mismatches
The first step that we can take is to perform a merger-only analysis by computing mismatches between the GRwaveform and the EDGB waveform using the formulaein Sec. A. This involves restricting to a given time (orfrequency) range over which to compute the mismatch.When performing tests of general relativity, LIGO per-forms such merger-only calculations. In [1], the authorsperformed an inspiral-merger-ringdown consistency testfor GW150914 by inferring final mass and spin parametersusing GR waveforms from the post-inspiral portion of thewaveform only, from the inspiral portion of the waveformonly, and comparing the resulting posterior distributionto that from the full waveform analysis. For GW150914,the merger-ringdown region was chosen to be [132 , Hz. In this region, the signal had a signal to noise ratio(SNR) of 16, which is larger than the full-waveform SNRof the other nine BBH detections in GWTC-1 [15].We thus compute mismatches between the GR andEDGB merger waveforms, shown in Fig. 5. We show themismatch (cf. Sec. A) for various values of √ α GB /GM (cf. Fig. 2). In particular, for a mismatch, we find √ α GB /GM (cid:46) . . For GW150914, we choose M ∼ M (cid:12) [14], and thus compute √ α GB (cid:46) km. Notethat though we shift the waveforms in time and phaseto compute a minimum mismatch, we do not vary theGR waveform parameters (mass and spin). Thus ourmismatch estimate is optimistic, and performing a full
150 175 200 225 250Inspiral Length [ M ]050001000015000 P e a k A m p li t ud e Ψ ( ) ( , ) Quadratic fitLinear fit
FIG. 4. Peak amplitude of the EDGB correction to the gravi-tational waveform as a function of inspiral length. We showthe length relative to the peak of the waveform (as in Fig. 3).The dashed black vertical line corresponds to the length ofthe EDGB merger simulation we present in this paper. Thepeak amplitude serves as a measure of the amount of seculargrowth in the waveform (cf. Fig. 3). We see that the seculargrowth attains a quadratic minimum, and thus for a shortenough inspiral length, we can obtain an EDGB gravitationalwaveform with minimal secular contamination. parameter-estimation analysis on our EDGB waveform isthe subject of future research.For heavier BBH systems, such as GW170729 with M = 84 M (cid:12) , which had 3 cycles in the LIGO band [15], wecan in theory use only the merger-ringdown EDGB wave-forms from numerical relativity simulations for data anal-ysis, without requiring EDGB inspiral waveforms. Note,however, that with all other parameters held equal, thislead to a lower constraint on √ α GB from the larger totalmass. Moreover, GW170729 has an SNR of ∼ , whichis less than the merger SNR of 16 for GW150914. Notethat the LIGO intermediate mass black hole search [36]which looked for BBHs with M ∈ [120 , M (cid:12) did notdetect any signals. B. Including inspiral
How much more could we gain if we additionally in-cluded the inspiral phase? Gaining access to the inspiralphase for EDGB waveforms is ongoing work, througheither implementing a renormalization scheme to removesecular effects as outlined in [18], or by stitching on post-Newtonian or parametrized post-Einstenian (ppE) EDGBwaveforms for the inspiral [27, 37], to obtain a full wave-form.In [3], the authors use the ppE formalism to bound √ α GB with GW150914. Fig. 15 of [3] shows the upperbounds on √ α GB , including values of O (20 , km, but . . . . . . √ α GB for M = 68 M (cid:12) [km] 10 M i n i m u m M e r g e r S N R .
050 0 .
075 0 .
100 0 .
125 0 .
150 0 .
175 0 . √ α GB /GM − − − % M i s m a t c h f r o m M e r g e r Max allowed √ α GB /GM LIGO testing GR
FIG. 5. Mismatch between general relativity GW150914 wave-form (cf. Sec. III A) and the corresponding EDGB-correctedgravitational waveform, as defined in Eq. (A1). We show themismatch for our merger waveform as a function of the EDGBcoupling parameter, √ α GB /GM . We show the maximumallowed value of √ α GB /GM from the regime of validity (cf.Sec. III B) in dot-dashed gray. The dashed horizontal line cor-responds to the LIGO mismatch of from testing GR withGW150914 [1]. The top vertical axis corresponds to √ α GB computed from √ α GB /GM on the bottom axis assuming that M = 68 M (cid:12) for GW150914. this is very sensitive to the dimensionless spins of theblack holes, which are poorly constrained (cf. [14, 15]).Thus, the authors do not place an upper bound on √ α GB .In [38], the authors place an upper bound of √ α GB (cid:46) . km for GW150914 using a ppE analysis, which is higherthan our merger-only analysis bound. Including a mergerphase to these inspiral-only analyses can thus improvetheir bounds on √ α GB . C. Comparison to observational and projectedconstraints
Let us now compare the merger-analysis result of √ α GB (cid:46) km with observational and predicted ob-servational constraints in the literature. We summarizethese present constraints in Table I. Most notably, Witeket al. [12] estimate from their scalar field calculations thatfor a GW151226-like system [39], the constraint would be √ α GB (cid:46) . km. Note that this signal has ∼ cyclesin the LIGO band (compared to ∼ in the LIGO bandfor GW150914) [15], and thus the inspiral phase, whichis not included in our estimate, plays a greater role forthis system. Moreover, this esimate was performed with amass ratio of q ∼ and total mass ∼ M (cid:12) , which leadsto stronger beyond-GR effects due to the higher curvatureof the smaller object. Reference √ α GB bound Cassini Shapiro time delay [40] (cid:46) O (10 ) km X-ray binary orbital decay [41] (cid:46) kmCompact star stability [42] (cid:46) . kmLIGO SNR 30 detections [43] (cid:46) O (1 − kmEDGB scalar simulations for GW151226 [12] (cid:46) . kmGW150914 ppE [38] (cid:46) . kmTABLE I. Observed and projected bounds on the EDGBlengthscale from various studies. The first to rows (in bold),correspond to observed bounds, from the Cassini probe con-straints on Shapiro time delay and observations of X-ray bina-ries. Note that all bounds are given in terms of the conventionsin our action (cf. Eq. (1)), chosen to be consistent with [12]. V. CONCLUSION
We have produced the first astrophysically-relevant nu-merical relativity binary black hole gravitational waveformin Einstein dilaton Gauss-Bonnet gravity, a beyond-GRtheory of gravity. We have focused on a system withparameters consistent with GW150914, the loudest LIGOdetection thus far. This extends our previous work forproducing such a waveform for GW150914 in dynamicalChern-Simons gravity [18].In Sec. II, we laid out our order-reduction scheme, whichwe use to obtain a well-posed initial value formulationand produce the leading-order EDGB correction to thegravitational waveform. In Sec. III D, we showed theEDGB-corrected waveforms for a system consistent withGW150914 (cf. Sec. III A). We find that there is seculargrowth in the inspiral phase (Sec. III E), and thus presenta merger-ringdown waveform that is free of secular growth.We thus focus on a post-inspiral-only analysis, andcompute the mismatch between the (background) GRwaveform and the EDGB-corrected waveforms, finding abound on the EDGB coupling parameter of √ α GB /GM (cid:46) km. This is a stronger result than inspiral-only analysesfor GW150914, which bound √ α GB (cid:46) . km. Notethat GW150914 has an SNR of 16 in the post-inspiralphase (cf. [1]), which is larger than the total SNR of eachother event in GWTC-1 [15]. Stitching on a parametrizedpost-Einstenian EDGB inspiral or removing the inspiralsecular growth from our simulations (cf. [18]) to take fulladvantage of an inspiral-merger-ringdown analysis is thesubject of future work.Our ultimate goal is to make these beyond-GR wave-forms useful for LIGO and Virgo tests of general rela-tivity [1, 2]. We can improve the mismatch analysis byallowing the GR waveform parameters to vary, thus check-ing for degeneracies in the GR-EDGB parameter space.Moreover, we can perform a more quantitative analysis byinjecting our beyond-GR waveforms into LIGO noise andcomputing posteriors recovered using present LIGO pa-rameter estimation and testing-GR methods [2, 14, 44, 45].Ultimately, we would like to generate enough beyond-GREDGB waveforms to fill the BBH parameter space. Wecan then produce a beyond-GR surrogate model [46] andperform model-dependent tests of GR. ACKNOWLEDGEMENTS
We thank Leo Stein, Helvi Witek, and Paolo Pani foruseful discussions. The Flatiron Institute is supported bythe Simons Foundation. Computations were performedusing the Spectral Einstein Code [30]. All computationswere performed on the Wheeler cluster at Caltech, whichis supported by the Sherman Fairchild Foundation andby Caltech.
Appendix A: Mismatches
Given the GR and EDGB-corrected waveforms (asshown in Fig. 2), let us consider the mismatch betweenthese waveforms. A more involved calculation wouldinvolve computing a mismatch in the presence of grav-itational wave detector noise and considering a rangeof parameters for the GR waveform to test for degenera-cies [47]. Here, we perform a simpler mismatch calculationbetween the background GR waveform Ψ (0)4 and the cor- responding EDGB-modified waveform considered in thisstudy (cf. Sec. III A). Once we have the EDGB correction Ψ (2)4 from the numerical relativity simulation, we intro-duce a coupling parameter √ α GB /GM before adding itto the GR waveform using Eq. (13) to obtain Ψ ( √ α GB ) .We then compute the mismatch as (cf. [48]) Mismatch( √ α GB ) ≡ (A1) − Re (cid:104) Ψ (0)4 , Ψ ( √ α GB ) (cid:105) (cid:113) (cid:104) Ψ (0)4 , Ψ (0)4 (cid:105) × (cid:104) Ψ ( √ α GB ) , Ψ ( √ α GB ) (cid:105) , where we have explicitly shown the dependence on √ α GB .We define the inner product (cid:104) , (cid:105) between two waveformsas (cid:104) Ψ , Ψ (cid:105) ≡ (cid:90) t end t start Ψ ( t ) [2] ˜Ψ ∗ ( t ) [1] dt , (A2)where ∗ denotes complex conjugation. This is preciselythe inner product used in [48]. We choose t start to bethe section of the waveform where EDGB effects are fullyramped-on, and choose t end to be the end of the numericalwaveform. This is equivalent, by Parseval’s theorem, toa noise-weighted inner product in the frequency domainwith noise power spectral density S n ( | f | ) = 1 . We shiftthe waveforms in time and phase when computing thisoverlap. [1] B. P. Abbott et al. (Virgo, LIGO Scientific), Phys.Rev. Lett. , 221101 (2016), [Erratum: Phys. Rev.Lett.121,no.12,129902(2018)], arXiv:1602.03841 [gr-qc].[2] B. P. Abbott et al. (LIGO Scientific, Virgo), (2019),arXiv:1903.04467 [gr-qc].[3] N. Yunes, K. Yagi, and F. Pretorius, Phys. Rev. D94 ,084002 (2016), arXiv:1603.08955 [gr-qc].[4] P. Kanti, N. E. Mavromatos, J. Rizos, K. Tamvakis, andE. Winstanley, Phys. Rev.
D54 , 5049 (1996), arXiv:hep-th/9511071 [hep-th].[5] D. J. Gross and J. H. Sloan, Nuclear Physics B , 41(1987).[6] F. Moura and R. Schiappa, Class. Quant. Grav. , 361(2007), arXiv:hep-th/0605001 [hep-th].[7] E. Berti et al. , Class. Quant. Grav. , 243001 (2015),arXiv:1501.07274 [gr-qc].[8] G. Papallo and H. S. Reall, Phys. Rev. D96 , 044019(2017), arXiv:1705.04370 [gr-qc].[9] G. Papallo, Phys. Rev.
D96 , 124036 (2017),arXiv:1710.10155 [gr-qc].[10] J. L. Ripley and F. Pretorius, Phys. Rev.
D99 , 084014(2019), arXiv:1902.01468 [gr-qc].[11] J. L. Ripley and F. Pretorius, Class. Quant. Grav. ,134001 (2019), arXiv:1903.07543 [gr-qc].[12] H. Witek, L. Gualtieri, P. Pani, and T. P. Sotiriou,(2018), arXiv:1810.05177 [gr-qc].[13] B. P. Abbott et al. (Virgo, LIGO Scientific), Phys. Rev.Lett. , 061102 (2016), arXiv:1602.03837 [gr-qc]. [14] B. P. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev.Lett. , 241102 (2016), arXiv:1602.03840 [gr-qc].[15] B. P. Abbott et al. (LIGO Scientific, Virgo), (2018),arXiv:1811.12907 [astro-ph.HE].[16] M. Isi, M. Giesler, W. M. Farr, M. A. Scheel, andS. A. Teukolsky, Phys. Rev. Lett. , 111102 (2019),arXiv:1905.00869 [gr-qc].[17] R. Nair, S. Perkins, H. O. Silva, and N. Yunes, (2019),arXiv:1905.00870 [gr-qc].[18] M. Okounkova, L. C. Stein, J. Moxon, M. A. Scheel, andS. A. Teukolsky, (2019), arXiv:1911.02588 [gr-qc].[19] S. Alexander and N. Yunes, Phys. Rept. , 1 (2009),arXiv:0907.2562 [hep-th].[20] M. B. Green and J. H. Schwarz, Phys. Lett. , 117(1984).[21] V. Taveras and N. Yunes, Phys. Rev. D78 , 064070 (2008),arXiv:0807.2652 [gr-qc].[22] S. Mercuri and V. Taveras, Phys. Rev.
D80 , 104007(2009), arXiv:0903.4407 [gr-qc].[23] M. Boyle and A. H. Mroue, Phys. Rev.
D80 , 124045(2009), arXiv:0905.3177 [gr-qc].[24] M. Okounkova, L. C. Stein, M. A. Scheel, and D. A. Hem-berger, Phys. Rev.
D96 , 044020 (2017), arXiv:1705.07924[gr-qc].[25] M. Okounkova, M. A. Scheel, and S. A. Teukolsky, Phys.Rev.
D99 , 044019 (2019), arXiv:1811.10713 [gr-qc].[26] M. Okounkova, L. C. Stein, M. A. Scheel, and S. A.Teukolsky, Phys. Rev. D , 104026 (2019). [27] K. Yagi, L. C. Stein, N. Yunes, and T. Tanaka,Phys. Rev.
D85 , 064022 (2012), [Erratum: Phys.Rev.D93,no.2,029902(2016)], arXiv:1110.5950 [gr-qc].[28] C. M. Bender and S. A. Orszag,
Advanced mathematicalmethods for scientists and engineers (McGraw-Hill BookCo., New York, 1978) pp. xiv+593, international Seriesin Pure and Applied Mathematics.[29] T. Hinderer and E. E. Flanagan, Phys. Rev.
D78 , 064028(2008), arXiv:0805.3337 [gr-qc].[30] “The Spectral Einstein Code (SpEC),” .[31] P. Kumar, J. Blackman, S. E. Field, M. Scheel, C. R.Galley, M. Boyle, L. E. Kidder, H. P. Pfeiffer, B. Szilagyi,and S. A. Teukolsky, (2018), arXiv:1808.08004 [gr-qc].[32] “Sxs gravitational waveform database,” .[33] G. Lovelace et al. , Class. Quant. Grav. , 244002 (2016),arXiv:1607.05377 [gr-qc].[34] S. Bhagwat, M. Okounkova, S. W. Ballmer, D. A. Brown,M. Giesler, M. A. Scheel, and S. A. Teukolsky, Phys. Rev. D97 , 104065 (2018), arXiv:1711.00926 [gr-qc].[35] M. Giesler, M. Isi, M. Scheel, and S. Teukolsky, (2019),arXiv:1903.08284 [gr-qc].[36] B. P. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev.
D100 , 064064 (2019), arXiv:1906.08000 [gr-qc]. [37] S. Tahura and K. Yagi, Phys. Rev.
D98 , 084042 (2018),arXiv:1809.00259 [gr-qc].[38] S. Tahura, K. Yagi, and Z. Carson, Phys. Rev.
D100 ,104001 (2019), arXiv:1907.10059 [gr-qc].[39] B. P. Abbott et al. (Virgo, LIGO Scientific), Phys. Rev.Lett. , 241103 (2016), arXiv:1606.04855 [gr-qc].[40] B. Bertotti, L. Iess, and P. Tortora, Nature (London) , 374 (2003).[41] K. Yagi, Phys. Rev.
D86 , 081504 (2012), arXiv:1204.4524[gr-qc].[42] P. Pani, E. Berti, V. Cardoso, and J. Read, Phys. Rev.
D84 , 104035 (2011), arXiv:1109.0928 [gr-qc].[43] L. C. Stein and K. Yagi, Phys. Rev.
D89 , 044026 (2014),arXiv:1310.6743 [gr-qc].[44] J. Aasi et al. (LIGO Scientific, VIRGO), Phys. Rev.
D88 ,062001 (2013), arXiv:1304.1775 [gr-qc].[45] N. J. Cornish and T. B. Littenberg, Class. Quant. Grav. , 135012 (2015), arXiv:1410.3835 [gr-qc].[46] V. Varma, S. E. Field, M. A. Scheel, J. Blackman,D. Gerosa, L. C. Stein, L. E. Kidder, and H. P. Pfeiffer,Phys. Rev. Research. , 033015 (2019), arXiv:1905.09300[gr-qc].[47] K. Chatziioannou, A. Klein, N. Yunes, and N. Cornish,Phys. Rev. D95 , 104004 (2017), arXiv:1703.03967 [gr-qc].[48] M. Boyle et al. , Class. Quant. Grav.36